COU11SE  OF  MATHEMATICAL  W011KS, 

BY  GEORGE  R.  PERKINS,  A.  My, 

Professor  of  Mathematics  and  Principal  of  the  State  Normal  School 

I.     PRIMARY  ARITHMETIC.     Price  21  cts. 


A  want,  with  young  pupils,  of  rapidity  and  accuracy  in  performing  operations  upon  wn'  ei 
.lasers  ;  a-n  imperfect  knowledge  of  Numeration  ;  Inadequate  conceptions  of  the  nature  i  <« 
(e.tttions  of  Fractions,  and  a  lack  of  familiarity  with  the  principles  of  Decimals,  have  indu  •*" 


thf  author  10  prepare  the  PRIMARY  ARITHMETIC. 

The  firs',  part  is  devoted  to  MENTAL  EXEROISES  and  the  second  to  Exercises  on  the  S  v 
«n<7  Blackboard. 

While  the  minds  of  young  pupils  are  disciplined  by  mental  exercises  (if  not  wearisome 
prolonged),  they  fail,  in  general,  in  trusting  to  "head-work"  for  their  calculation?;  and  in  t> 
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of  early  familiarity  with  such  processes:  these  considerations  have  induced  the  Author  to  dev  »< 
oart  of  his  book  to  primary  written  exercises. 

It  has  been  received  with  more  popularity  than  any  Arithmetic  heit-iofore  issued. 

II.    ELEMENTARY  ARITHMETIC.    Price  42  cts. 

Has  recently  been  carefully  revised  and  enlarged.    It  will  be  found  concise,  yet  lucid,     ftieac).*' 
the  radical  relations  of  numbers,  and  presents  fundamental  principles  in  analysis  and  examp  ei 
)  It  leaves  nothing  obscure,  yet  it  does  not  embarrass  by  multiplied  processes,  nor  er.feeble  ->i 
'  minute  details. 

In  this  work  all  of  the  examples  or  problems  are  strictly  practical,  made  up  as  they  are  i  ; 
great  measure  of  important  statistics  and  valuable  facjs  in  history  and  f  hilosophy,  which  i  i> 
thus  unconsciously  learned  in  acquiring  a  knowledge  of  the  Arithmetic. 

Fractions  are  placed  immediately  after  Division  ;  Federal  Money  is  treated  as  and  with  lii 
cim?l  Fractions;  Proportion  is  placed  before  Fellowship.  Alligation,  and  such  rules  as  requ  t- 
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swers  to  all  of  the  examples  are  given. 

The  work  will  be  found  -to  be  an  improvement  on  most,  if  not  all,  previous  element? 
Arithmetics  in  the  treatment  of  Fractions,  Denominate  Numbers,  Rule  of  Three,  Interest,  E  ji  .. 
ticn  of  Payments,  Extraction  of  Roots,  and  many  other  subjects. 

Wherever  this  work  is  presented,  the  publishers  have  heard  but  one  opinion  in  regard  10  1 
merits,  and  that  most  favorable. 

III.    HIGHER  ARITHMETIC.    Price  84  cts. 

I  Tne  present  edition  has  been  revised,  many  subjects  rewritten,  and  much  new  matter  adile 
and  cotttams  an  APPENDIX  of  about  GO  pages,  in  which  the  philosophy  of  the  more  dime1.1 
operai  ions  and  interesting  properties  of  numbers  are  fully  discussed.  The  work  is  what  its  nai  ••• 
jpurpoits,  a  Higher  Ar5thmetic,  and  will  be  found  to  contain  many  entirely  new  principles  whi  l 
have  never  before  ap^ared  in  any  Arithmetic.  It  has  received  the  strongest  recom  nendafir  ••> 
from  hrndreds  of  the  best  teachers  the  country  affords. 

IV.  ELEMENTS  OF  ALGEBRA.    Price  84  cts. 

This  work  is  an  introduction  to  the  Author's  "  Treatise  on  Algebra,"  and  is  designed  esj  * 
dally  for  the  use  of  Common  Schools,  and  universally  pronounced  "admirably  adapted  loth. 
purpose." 

V.  TREATISE  ON  ALGEBRA.     Price  SI  50. 

This  work  contains  the  higher  parts  of  Algebra  usually  taught  in  Colleges  ;  a  new  methyl 
of  cute:  and  higher  equation?  as  veil  as  the  THEOREM  OF  STURM,  by  which  we  may  at  one* 
determine  the  numbei  of  real  roots  of  any  Algebraic  Equation,  wiih  much  moie  ease  than  bf 
previously  discovered  method. 

In  the  present  revised  editiyn,  one  entire  chapter  on  the  subject  of  COMTI*UBD  FRA.CTIC  «.« 
ha*  been  added. 

VI.    ELEMENTS  OF  GEOMETRY,  WITH  PRACTICAL  APPLICATIONS.    SI 

The  author  has  added  throughout  the  entire  Work,  PRACTICAL  APPLICATIONS,  which,  in  th< 
Mimiition  of  Teachers,  is  an  important  consideration. 

An  eminent  Professor  of  Mathematics,  in  speaking  of  this  work,  says:  "  We  hav,  uioj  tetl 
it,  because  it  follows  more  closely  the  best  model  of  pure  <reomeirical  reasoning,  which  "er  ha* 
oeen.  and  perhaps  ever  will  be  exhibiud  ;  and  becay.se  the  A 


.         e  Author  h;^  cwtowe*/  x>n  •  of  the 

Unportant  principles  of  the  great  master  of  Geometricians,  arid  more  especia,.r  ha.*  sh.iwn  that 
iis  theorems  are  not  mere  theory,  iy  many  practical  application*  :  a  quality  in  a  text-book  »' 
'his  •v.icnce.  i:'  less  uncommon  than  u  is  important." 

6 


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original,  scientific  and  practical,  and  destined  wherever  it  is  introduced  to  supersede  at  once  all 


PERKINS'    SERIES. 


THE 


PRACTICAL  ARITHMETIC: 


DESIGNED  FOR  SUCH 

INSTITUTIONS 

A3   REQUIRE  A  GREATER  NUMBER  OF  EXAMPLES  THAN  ARE 
GIVEN  IN  THE 

ELEMENTARY  ARITHMETIC. 

BY 

GEORGE      R.     PERKINS,    A.M. 

PRINCIPAL  AND   PROFESSOR   OF   MATHEMATICS    IN   NEW  YORK    STATE    NORMAl 

SCHOOL,   AUTHOR   OF   "ELEMENTARY  ARITHMETIC,"   "HIGHER 

ARITHMETIC,"   "  ELEMENTS    OF   ALGEBRA," 

ETC.    ETC. 


NEW  YORK: 

D.  APPLETON  &  COMPANY,  200  BROADWAY  ' 

M  DCCC  LII. 


Entered  according  to  Act  of  Congress,  in  Ine  yoar  U5i 
BY  GEORGE  R.  PERKINS, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Northern 
District  of  New  York. 


PREFACE. 


WHILE  many  distinguished  teachers  unite  in  pronouncing 
my  Elementary  Arithmetic  as  the  best  work  of  the  kind, 
well  adapted  to  the  purpose  of  teaching  the  science,  as  well 
as  the  art,  of  Arithmetic ;  others  have  often  expressed  to 
me  their  belief  that  its  usefulness  would  be  greatly  in- 
creased by  the  addition  of  more  examples  of  a  practical 
kind ;  and  in  many  cases  I  have  been  strongly  urged  to 
omit.the  answers. 

I  am  convinced  that  for  certain  grades  of  institutions  it 
would  be  well  to  have  a  greater  variety  of  examples  for 
practice,  but  I  do  not  so  readily  see  the  advantage  of  omit- 
ting the  answers.  I  have  always  been  inclined  to  believe 
the  omission  of  the  answers  gave  an  opportunity  for  the 
pupil,  and  in  some  cases  for  the  teacher,  to  pass  over  many 
principles  without  thoroughly  understanding  them,  since  a 
result  would  frequently  be  obtained  which  might  perhaps 
be  quite  erroneous ;  and  having  no  answer  with  which  to 
compare  it,  he  dismisses  the  subject  with  the  belief  that  he 
has  conquered  the  difficulty,  and  that  he  understands  clearly 
the  principle  especially  designed  to  be  brought  out  by  the 
example. 

With  these  facts  in  view,  I  have  prepared  the  Practical 
1* 


VI  PREFACE. 


Arithmetic,  which  is  designed  not  to  supply  the  place  of 
the  Elementary  Arithmetic,  but  is  designed  for  the  use  of 
such  institutions  as  require  a  greater  number  of  examples 
than  are  given  in  that  work. 

In  many  cases  the  rules  have  been  rewritten  and  con- 
densed, so  as  to  bring  the  same  rule  to  apply  to  as  large  a 
class  of  operations  as  possible. 

The  arrangement  of  the  subjects  is  in  many  respects  dif- 
ferent from  that  given  in  the  Elementary  Arithmetic.  The 
whole  is  divided  into  chapters,  and  under  each  chapter  the 
examples  begin  a  new  numbering,  so  that  under  the  same 
chapter  there  are  no  two  examples  of  the  same  number. 

In  forming  these  examples,  great  care  has  been  taken  to 
make  them  as  practical  as  possible.  The  answers  have  been 
omitted  in  the  body  of  the  work,  but  they  are  given  in.  the 
Appendix,  where  they  are  arranged  in  reference  to  the 
chapter  and  section. 

The  questions  designed  to  test  the  pupil's  knowledge  of 
the  principles  and  rules,  are  also  given  in  the  Appendix. 
In  the  construction  of  these  test  questions,  care  has  been 
taken  to  draw  out  the  actual  knowledge  of  the  pupil,  as  in 
many  cases  they  cannot  be  correctly  answered  without  a 
thorough  knowledge  of  the  subject.  It  is  believed  these 
questions  will  be  found  of  valuable  assistance  to  the  teacher 
in  reviews. 

GEO.  R.  PERKINS. 

ALBANY,  JUNE,  1851. 


CONTENTS. 


CHAPTER  I. 

Page 

Arithmetic  defined 1 


CHAPTER  II. 

Notation 2 

Roman  Notation 2 

Arabic  Notation 4 

Numeration 9 

CHAPTER  III. 

Addition 11 

Proof  of  Addition 14 

CHAPTER  IV. 

Subtraction 20 

Proof  of  Subtraction 24 

Questions  involving  Addition  and  Subtraction 26 

CHAPTER  V. 

Multiplication  27 

Proof  of  Multiplication 33 

Questions  involving  Addition,  Subtraction,  and  Multiplication...  35 

CHAPTER  VI. 

Division 38 

Proof  of  Division 43 

Questions  involving  the  four  ground  rules ...".. 47 


Vlll  CONTENTS. 

CHAPTER  VII. 

Page 

General  Problems  and  Principles 50 

Principles  evolved  from  Division 53 

CHAPTER  VIII. 

Prime  Numbers 54 

Greatest  Common  Divisor 57 

Least  Common  Multiple  60 

Cancelation 63 

CHAPTER  IX. 

Fractions  defined  65 

Reduction  of  Fractions , 67 

Addition  of  Fractions 74 

Subtraction  of  Fractions 75 

Multiplication  of  Fractions 76 

Division  of  Fractions 77 

Reciprocals 78 

Miscellaneous  Examples  in  Common  Fractions 79 

CHAPTER  X. 

Decimal  Fractions 82 

Addition  of  Decimals 86 

Subtraction  of  Decimals 87 

Multiplication  of  Decimals 88 

Division  of  Decimals 90 

Promiscuous  Examples  in  Decimals 93 

Reduction  of  Common  Fractions  to  Decimals 95 

Reduction  of  Decimals  to  Common  Fractions 97 

Federal  Money  98 

Promiscuous  Examples  in  Federal  Money 101        $ 

Abridged  Method  for  operations  in  Federal  Money 107 

CHAPTER  XI. 

Denominate  Numbers. 109 

English  or  Sterling  Money 110 


CONTENTS.  IX 

Page 
Troy  Weight  ................................................................  Ill 

Apothecaries'  Weight  ......................................................  Ill 

Avoirdupois  Weight  .......................................................  112 

Long  Measure  ...................................................  ...........  112 

Cloth  Measure  ...............................................................  113 

Square  Measure  ............................................................  114 

Solid  or  Cubic  Measure  ...................................................   115 

Wine  Measure  ...............................................................  116 

Ale  or  Beer  Measure  ......................................................  116 

Dry  Measure  ................................................................  116 

Time  ...........................................................................  117 

Circular  Measure  ............................................................  118 

Table  of  particular  Weights  and  Measures  ..........................  119 

Reduction  of  Denominate  Quantities  ..................................  123 

Promiscuous  Exercises  in  Reduction  of  Denominate  Quantities...   127 
Addition  of  Denominate  Numbers  .....................................  132 

Subtraction  of  Denominate  Numbers  ..................................  135 

Exercises  in  Addition  and  Subtraction  .................................  137 

Multiplication  of  Denominate  Numbers  ...............................  139 

Division  of  Denominate  Numbers  ......................................  .  142 

Questions  involving  the  four  preceding  rules  ........................  143 

Duodecimals  ................................................................  146 

Addition  and  Subtraction  of  Duodecimals  ............................  146 

Multiplication  of  Duodecimals  ...........................................  147 

Division  of  Duodecimals  ...................................................  150 

Addition  of  Denominate  Fractions  .....................................  152 

Subtraction  of  Denominate  Fractions  ..................................  153 

Exercises  in  Denominate  Fractions  .....................................  154 


CHAPTER  XII. 
Percentage  ..................................................................  156 

Commission,  Brokerage,  and  Stocks  ....................................  159 

Assessment  of  Taxes  ......................................................  162 

Custom-house  Business  ....................................................  164 

Insurance  .....................................................................   167 

Profit  and  Loss  .............................................................  169 

Simple  Interest  ............................................................  173 

Interest  when  the  time  is  estimated  in  days  ........................  179 


X  CONTENTS. 

Page 

Partial  Payments 181 

Problems  in  Interest 186 

Discount 191 

Compound  Interest 192 

Banking 193 

Bank  Discount 194 

CHAPTER  XIII. 

Analysis 198 

Ratio  200 

Practice  205 

Reduction  of  Currencies 207 

CHAPTER  XIV. 

Proportion 214 

Single  Rule  of  Three,  first  form 216 

Single  Rule  of  Three,  second  form 220 

Compound  Proportion  224 

Rule  for  Compound  Proportion 226 

Arbitration  of  Exchange 229 

Partnership  or  Fellowship 231 

Double  Fellowship 233 

CHAPTER  XY. 

Average 235 

Equation  of  Payments 238 

Rule  for  finding  Cash  Balance  of  Book  Account v 242 

Alligation  Medial 246 

Alligation  Alternate 248 

CHAPTER  XVI. 

Involution  ,. 253 

Evolution iy. 254 

Rule  for  extracting  the  Square  Root 260 

Examples  involving  the  principles  of  the  Square  Root...-, 262 

Demonstration  of  Rule  for  extracting  Cube  Root * 268 

Examples  involving  the  principles  of  the  Cube  Root 275 


CONTENTS.  XI 

CHAPTER  XVII. 

Page 

Arithmetical  Progression 277 

Geometrical  Progression  280 

CHAPTER  XVIII. 

Mensuration 284 

Promiscuous  Questions  .j 294 

APPENDIX..  ,.  305 


THE 

PRACTICAL  ARITHMETIC, 


CHAPTER   I. 

ARITHMETIC. 

§  1.  EVERY  single  thing  is  called  a  unit,  A  NUMBER,  then, 
must  represent  one  or  more  units. 

If  the  units  represented  by  a  number  have  no  reference 
to  particular  things,  the  number  is  called  an  abstract  number. 

Thus,  the  number  Eight  is  an  abstract  number,  because 
it  does  not  mean  eight  apples,  or  eight  dollars,  or  eight  any 
particular  things,  but,  simply,  eight. 

If  the  units  represented  by  a  number  have  reference  to 
particular  things,  the  number  is  called  a  concrete  or  denom- 
inate number. 

Thus,  the  number  Eight,  meaning  eight  apples,  or  eight 
dollars,  or  eight  any  particular  things,  is  a  denominate  num- 
ber. 

NOTE. — Units  or  numbers  are  called  denominate,  because  they  de- 
nominate or  name  particular  things. 

$  2.  ARITHMETIC  treats  of  numbers,  whether  abstract  or 
denominate.  As  a  science,  it  is  a  knowledge  of  the  proper- 
ties and  relations  of  numbers ;  as  an  art,  it  is  a  practical 
facility  in  computing  by  numbers 


NOTATION. 


[CHAP.  n. 


Numbers^  may  be  expressed  by  words,  by  letters,  or  by 
figures.  These  two  latter  methods  form  different  systems 
of  notation. 


CHAPTER   II. 

NOTATION. 

§  3.  NOTATION  is  the  expressing  of  numbers  by  letters 
or  figures. 

The  use  of  letters  for  this  purpose  was  adopted  by  several  early 
nations,  especially  by  the  Romans  ;  hence  called  the 

ROMAN  METHOD. 

This  method  employed  seven  capital  letters,  viz. :  I  for  one ; 
V  for  five  ;  X  for  ten  ;  L  forffty;  C  for  a  hundred  ;  D  for  Jive  hun- 
dred; M  for  a  thousand.  By  means  of  these  letters,  repeated  or 
variously  combined,  any  number  may  be  expressed  ;  thus  : 


I  stands 

for  One. 

XV  stands  for  Fifteen. 

II      " 

"  Two. 

XVI    " 

"   Sixteen. 

III      « 

«  Three. 

XVII    " 

"   Seventeen. 

IV      " 

V      « 

"  Four. 
"  Five. 

XVIII    " 
XIX    " 

"  Eighteen. 
"  Nineteen. 

VI      " 
VII     " 
VIII     " 

IX      « 

«  Six. 
"  Seven. 
"  Eight. 
"  Nine. 

XX    « 
XXI    " 
L    " 

C    " 

"   Twenty. 
"   Twenty-one. 
"   Fifty. 
"   One  hundred. 

"X"       " 

«  Ten. 

D    « 

"  Five  hundred. 

XI     " 

"  Eleven. 

DO    « 

"   Six  hundred. 

XII     « 

"  Twelve. 

M    " 

"   One  thousand. 

XIII      « 

"  Thirteen. 

MM    « 

"  Two  thousand. 

XIV      « 

"  Fourteen. 

MMM    « 

"   Three  thousand. 

A  letter  placed  before  another  letter  of  greater  value,  takes  away 
its  own  value  from  the  greater  ;  as  V,  five  ;  IV,  one  from  five,  or  four. 

A  letter  placed  after  another  letter  of  greater  value,  adds  its  own 
value  to  the  greater ;  as  V,  five  ;  VI,  five  and  one,  or  six. 


§  3.]  ROMAN  METHOD.  3 

The  repeating  of  a  letter  repeats  the  value  of  the  letter. 

A  horizontal  line  over  a  letter  increases  it  a  thousand-fold ;  thus, 

D,  five  hundred  ;  D,  five  hundred  thousand. 

1-5.  Write,  after  the  Roman  method,  seventeen ;  forty- 
two  ;  twenty-six ;  ninety-eight ;  one  hundred  and  three. 

6—11.  Write  eighty-two ;  fifty-seven  ;  seventy-nine;  four 
hundred  and  thirty  ;  six  hundred  and  eighty  ;  two  thousand 
and  seven. 

12-14.  Write  three  hundred  thousand  ;  nine  hundred  and 
sixty  thousand  ;  one  million. 

15.  How  many  dollars  are  represented  by  bank  bills, 
marked  as  follows  :  X  ;  XX  ;  V  ;  I  ;  II ;  III  ? 

16.  How  many  dollars  are    represented   by  bank  bills, 
marked  X  ;  C  ;  V  ;  L  ;  I  ? 

17-28.  Read  the  following  numbers  :  XXVI;  CXLIY ; 
XCVIII ;  MCCCXII ;  MDCCCL  ;  MDCCCCLXXII ; 

D;    DMMDCCLXX;    M;     XXCIV ;     MDCLXXXVIII ; 
MDCCLXXV. 

Roman  numbers  are  now  used  chiefly  to  mark  volumes, 
chapters,  or  lessons  ;  to  indicate  the  hours  on  clock  or  watch 
faces  ;  for  dates  upon  tombstones,  tablets,  &c. ;  and  to  des- 
ignate the  year  of  the  Christian  era. 

NOTE. — Among  the  Romans  each  I  represented  a  finger ;  the 
whole  hand  spread  out,  thus,  \\JA  represented  fve.  This  character 
was  afterwards  written  V.  X,  ten,  is  merely  two  fives  written  one 
above  the  other ;  thus,  (V.) 

C,  the  initial  of  Centum,  the  Latin  for  one  hundred,  was  often 
written  C  ;  this  character  cut  in  two,  will  leave  L,  fifty,  for  its 
lower  half. 

M,  the  initial  of  Mille,  the  Latin  for  one  thousand,  was  originally 
written  CIO.  The  right-hand  portion  of  this  character  is  10,  which 
represents  five  hundred,  and  is  now  expressed  by  D. 


NOTATION.  [CHAP.  n. 


ARABIC    NOTATION. 

§  4.  The  method  of  notation  in  common  use  is  the  Ar.a,bic. 
This  method  employs  ten  characters,  viz. : 

1,     2,     3,     4,     5,     6,     7,     8,     9,     0. 

One,      Two,    Three,  Four,    Five,     Six,    Seven,  Eight,  Nine,  Naught. 

Each  of  these  characters,  except  the  Naught,  is  called  a 
digit  ;*  and  the  first  nine  taken  together  are  called  the  nine 
digits. 

The  digits  are  also  called  significant  figures. 

$  5.  The  significant  figures  have  unchanging  values  ;  that 
is,  they  always  represent  Units  or  Ones;  but  the  units 
which  they  represent  differ  in  value. 

If  a  significant  figure  stand  disconnected  from  other  fig- 
ures, the  value  of  its  unit  is  called  its  simple  value.  Thus, 
6  means  six  units  of  simple  value. 

But  if  a  significant  figure  stand  in  connection  with  other 
figures,  the  value  of  its  unit  is  called  its  local  value,  because 
this  value  depends  upon  the  place  which  such  figure  occu- 
pies, in  relation  to  the  figures  with  which  it  is  connected. 

Thus,  in  the  number  3456,  which  consists  of  four  signifi- 
cant figures  standing  in  connection  with  each  other,  each 
figure  expresses  units  ;  but  units  of  different  values. 

The  6  occupies  the  right-hand  or  first  place  ;  and  its  units 
are  said  to  be  of  the  first  order :  their  value  is  their  simple 
value  ;  that  is,  they  represent  six  single  ones. 

The  5  occupies  the  second  place  ;  and  its  units  are  said  to 
be  of  the  second  order  :  their  value  is  ten  times  greater  than 


*  From  the  Latin,  digitus,  a  finger ;  because  the  ancients  used  to  do  their  reck 
oning  on  their  fingers.    Originally  10,  ten,  was  also  called  a  digit. 


§  6.]  ARABIC  NOTATION.  5 

the  value  of  the  units  of  the  first  order ;  that  is,  they  repre- 
sent 5  tens. 

The  4  occupies  the  third  place  ;  and  its  units  are  said  to 
be  of  the  third  order  :  their  value  is  ten  times  greater  than 
the  value  of  the  units  of  the  second  order,  and  one  hundred 
times  greater  than  the  value  of  the  units  of  the  first  order ; 
that  is,  they  represent  4  hundreds. 

The  3  occupies  the  fourth  place ;  and  its  units  are  said  to 
be  of  the  fourth  order  :  their  value  is  ten  times  greater  than 
the  value  of  the  units  of  the  third  order,  and  one  thousand 
times  greater  than  the  value  of  the  units  of  the  first  order ; 
that  is,  they  represent  3  thousands. 

Each  unit,  then,  of  the  6,  represents  a  simple  unit :  each 
unit  of  the  5  represents  a  ten  :  each  unit  of  the  4  represents 
a  hundred :  each  unit  of  the  3  represents  a  thousand. 

So  we  might  proceed  with  any  number  of  figures  thus 
connected  together.  Hence  we  discover  this  fundamental 
property  : 

Every  fyure  in  a  number  represents  a  value  ten  times 
greater  than  that  of  the  fyure  next  to  it  at  its  right  hand. 

§  6.  The  naught,  0,  represents  the  absence  of  number. 
It  shows  that  in  the  place  which  it  occupies,  no  value  is  to 
be  expressed. 

Thus,  in  the  preceding  number  3456,  for  the  4  and  5, 
substitute  naughts,  thus,  3006.  These  naughts  show  that 
there  are  no  units  of  the  second  order,  or  tens,  and  no  units 
of  the  third  order,  or  hundreds,  in  the  number. 

Again,  7  standing  alone,  represents  seven  units  of  the  first 
order,  or  of  simple  value  :  a  naught  written  at  the  right  of 
the  7,  thus,  70,  shows  that  there  are  no  units  of  the  first 
order  in  the  number.  The  7  now  occupies  the  second  place, 
and  represents  units  of  the  second  order,  or  7  tens. 

1* 


6  NOTATION.  [CHAP.  n. 

Two  naughts  written  at  the  right  of  the  7,  thus  700, 
show  that  there  are  no  units  either  of  the  first  or  of  the  sec- 
ond order  in  the  number.  The  7  now  occupies  the  third 
place,  and  represents  units  of  the  third  order,  or  7  hundreds, 
and  so  on. 

Hence  this  property : 

Every  naught  at  the  right  of  a  significant  figure  increases 
the  value  which  that  figure  represents  ten-fold. 

And  conversely : 

Every  naught  removed  from  the  right  of  a  significant 
figure,  diminishes  the  value  which  that  figure  represents 
ten-fold. 

NOTE. — It  will  be  seen  that  the  office  of  the  naught  is  to  keep  the 
significant  figures  in  their  proper  places  ;  so  that  they  shall  correctly 
express  the  order  of  units  which  they  are  intended  to  represent. 
Annexing  them  to,  or  taking  them  from,  the  right  of  a  digit,  increases 
or  diminishes  its  value,  by  causing  that  digit  to  occupy  a  place  fur- 
ther to  the  left  or  further  to  the  right. 

§  7.  We  have  already  seen  that  the  first  place  of  a  num- 
ber is  occupied  by  units,  the  second  place  by  tens,  the  third 
place  by  hundreds,  and  the  fourth  place  by  thousands. 

In  writing  numbers,  then,  which  do  not  contain  more  than 
four  places,  the  pupil  will  probably  find  no  difficulty. 

Begin  with  the  units  of  the  highest  order  mentioned, 
and  in  whatever  place  no  units  are  required,  be  careful  to 
write  0. 

Express  in  figures  seven  thousand  nine  hundred  and  four. 

In  this  example  the  highest  order  of  units  is  thousands, 
of  which  there  are  7  :  the  next  order  is  hundreds,  of  which 
there  are  9  :  the  next  order  is  tens,  of  which  there  are  0 : 
the  next  order  is  simple  units,  of  which  there  are  4.  The 
whole  number  is  7904. 


§  8.]  ARABIC  NOTATION.  7 

29-43.  Express  in  figures  : 

Twenty.    (Two  tens  and  no  Eight  thousand  and  seven. 

units.)  Nine  thousand  and  twenty- 
Thirty-seven.  (Three  tens  and  seven. 

seven  units.)  Four  thousand  and  six. 

Ninety-eight.  Three  thousand. 

Three   hundred    and    thirty-  One  thousand  and  one. 

seven.  One  thousand  and  one  hun- 
Four  hundred  and  seven.  dred. 

Two  thousand  four  hundred  One  hundred  and  one. 

and  thirty-seven.  One  thousand  one  hundred 
Six  thousand  four  hundred  and  one. 

and  seven. 

§  8.  Suppose  it  be  required  to  express  in  figures  the  num- 
ber Sixty-one  millions,  nine  hundred  and  thirty-four  thou- 
sands, four  hundred  and  sixty-five. 

There  are  three  denominations  in  this  number :  Millions, 
Thousands,  and  Units. 

Write,  first,  the  figures  that  express  how  many  millions 
there  are  in  the  number:  61.  Write,  next,  the  figures  that 
express  how  many  thousands  there  are  in  the  number :  934. 
Write,  finally,  the  figures  that  express  how  many  units  there 
are  in  the  number :  405.  Connecting  these  groups  in  a  line, 

61,934,405, 

we  have  the  number  that  was  to  be  expressed. 

The  following  are  the  names  of  the  first  eight  groups,  or 
periods,  counting  from  the  right  towards  the  left :  Units, 
Thousands,  Millions,  Billions,  Trillions,  Quadrillions, 
<J H 'nit i Until*,  Scxtillions,  Sejitil/i(mx,  Octillions.  These  pe- 
riods may  be  extended  indefinitely. 

Each  period  contains  three  places.     In  the  first  or  right- 


8  NOTATION.  [CHAP.  n. 

hand  place,  the  units  of  the  period  must  be  written  :  in  the 
second  place,  the  tens  must  be  written :  in  the  third  place, 
the  hundreds  must  be  written. 

In  expressing  large  numbers,  then,  by  figures,  begin  with 
the  highest  denomination,  and  write  out  its  period  as  re- 
quired, and  so  proceed  with  all  the  periods  ;  taking  care  that 
each  period  except  the  left-hand  one,  has  its  three  places  oc- 
cupied  either  by  digits  or  by  naughts. 

Express  in  figures  the  number  Seven  trillions,  six  millions, 
and  thirty-one. 

This  number  involves  five  periods,  (although  but  three 'are  heard 
in  reading,)  Trillions,  Billions,  Millions,  Thousands,  and  Units.  In  the 
highest  or  trillions'  period,  the  unit's  place  only  is  occupied  :  7.  In 
the  next,  or  billions'  period,  there  are  no  hundreds,  tens,  or  units ; 
therefore  write  000.  In  the  millions'  period,  there  are  no  hundreds, 
no  tens,  but  simply  G  units  ;  tlterefore  write  for  this  period  006.  The 
thousands'  period  requires  no  digits  ;  therefore  write  for  that  period 
000.  In  the  units'  period  there  are  no  hundreds  to  be  expressed  ; 
therefore  write  0  in  the  hundreds'  place :  in  the  tens'  place  write  3  : 
in  the  units'  place  write  1. 

The  whole  number,  then,  is  7,000,006,000,031. 

The  following  will  exhibit  the  periods,  and  the  places  in  each 
period,  as  high  as  Octillions. 


a  |s|  1  §s  |  is  1 1||  i||  «|  1 1 1  |  i|  i  in  il 

3  2  1, 4  G  8,  9  0  7,8  4  3,  0  0 1,0  00,^7  4 1,8  01, 4^*2  21 
44—49.  Express  in  figures  the  following  numbers :  (two 


§  9.]  NUMEKATION.  9 

periods.)  Twenty-seven  thousand  three  hundred  ;  nine  hun- 
dred and  forty  thousand  and  two  hundred  ;  thirty-six  thou- 
sand four  hundred  and  fifty-six  ;  five  hundred  and  one  thou- 
sand ;  ninety-eight  thousand  ;  eleven  thousand. 

50-53.  (Three  periods.)  Forty-six  millions,  nine  hundred 
and  thirty  thousand,  six  hundred  and  fifty-nine  ;  three  hun- 
dred and  seven  millions,  eight  hundred  and  two  thousand, 
five  hundred  and  nine ;  nine  hundred  and  eighty-one  mil- 
lions, seven  hundred  ;  ten  millions,  ten  thousand  and  ten. 

54-57.  (Four  periods.)  Ninety-six  billions,  forty-eight 
millions,  seventy  three  thousand  and  ninety-eight ;  eight 
hundred  and  seven  billions  and  six  ;  ninety  billions  and  four 
thousand  and  ten ;  eight  hundred  billions,  six  millions  and 
seven. 

58-62.  (Five  periods.)  Forty-eight  trillions  ;  six  hundred 
and  nine  trillions ;  nine  hundred  and  eighty  trillions,  four 
billions  and  seven ;  three  trillions  and  two ;  nine  trillions 
and  two  thousand. 

63-68.  Thirty-six  sextillions  and  ninety-eight ;  four  quad- 
rillions, eight  trillions,  five  thousand  and  ninety-four  ;  thirty- 
five  quadrillions,  ninety-eight  billions  and  sixty-three ;  nine 
hundred  millions,  seven  hundred  thousand,  five  hundred  and 
ninety  ;  eighty-six  septillions  and  five  billions  ;  nine  hundred 
billions  and  forty-six  thousand. 

NOTE. — The  preceding  method  of  Notation  is  the  French,  which  ia 
now  in  almost  universal  use.  The  English  method  counts  six  places 
in  the  period. 

NUMERATION. 

§  9.  NUMERATION  is  the  reverse  process  of  Notation. 
Notation  is  the  method  of  expressing  in  figures  numbers 
that  are  written  in  words  ;  Numeration  is  the  method  of 
expressing  in  words,  numbers  that  are  written  in  figures. 


10  NOTATION.  [CHAP.  n. 

Notation  answers  to  writing,  and  Numeration  answers  to 
reading. 

To  read  large  numbers  with  facility,  separate  them,  as 
before,  into  periods  of  three  places  each,  counting  from  units. 
Then  commencing  at  the  left  hand,  read  the  figures  in  each 
period,  adding  the  name  of  the  period.  The  name  of  the 
unit's  period  need  not  be  added. 

Thus,  for  example,  read  the  following  :  37192854675. 
Separate  the  figures  into  periods  as  directed.  This  may  be 
done  in  the  imagination,  or  by  the  aid  of  the  comma.  The 
number  will  then  be  read, 

37  billions,  192  millions,  854  thousands,  675. 

69-75.  Read  the  following  numbers  :  678210  ;  5493678  ; 
456321980;  779146005;  42567000123901;  327980060; 
32987654300000*098. 

76-80.  563428670009;  358920761;  987678932; 
4560007980540068  ;  33492677005316896321. 

Let  the  following  sentences  be  read  : 

81.  The  distance  of  the  earth  from  the  nearest  fixed  star 
is  supposed  to  be  about  20000000000000  miles. 

82.  The  distance  of  the  moon  from  the  earth  is  236847 
miles. 

83.  The  planet  Mercury  is  distant  from  the  sun  36814721 
miles. 

84.  Yenus  is  distant  from  the  sun  68791752  miles. 

85.  Mars  is  144907630  miles  distant  from  the  sun. 

86.  Jupiter  is  494499108  miles  distant  from  the  sun. 

87.  The  diameter  of  the  sun  is  883246  miles. 

88.  The  circumference  is  2774799  miles. 


§  10.]  ADDITION.  11 

89.  The  square  miles  on  his  surface  is  2450830241208. 

90.  The  number  of  cubic  miles  is  360781001204398299. 

91.  The  amount  of  tea  consumed  in  the  United  States  for 
the  years  1842-1846,  was  73376290  pounds. 

92.  The  amount  of  coffee  consumed  for  the  same  time 
was  561707046  pounds. 

NOTE. — To  convey  some  idea  of  the  number  in  a  trillion,  it  may  be 
stated  that  not  a  trillion  seconds  have  eVpned  since  the  birth  of  Christ, 


CHAPTER    III. 

ADDITION. 

§  10.  ADDITION  is  the  process  of  uniting  two  or  more  num- 
bers so  as  to  form  one  number  ;  thus,  8  and  6  are  14. 

Numbers  of  the  same  ldnd  or  denomination  only  can  be 
added  ;  thus  we  may  add  8  oranges  to  6  oranges,  or  8  dol- 
lars to  6  dollars,  or,  simply,  8  (units)  to  6  (units)  ;  but  we 
cannot  add  8  oranges  to  6  dollars,  or  8  dollars  to  6  units. 

The  single  number,  formed  by  uniting  two  or  more  num- 
bers, is  called  the  SUM. 

The  sign  or  symbol  used  for  addition  is  +•  Thus,  6  +  8 
Kieans  6  added  to  8.  The  symbol  =,  which  is  not  confined 
to  addition,  means  equal  to;  thus,  6  +  8  =  14,  is  the  same 
as  if  written  6  added  to  8  is  equal  to  14.  The  expression 
be  read  6  and  8  are  14.* 


*  The  symbol  +  is  often  read  plus,  which  is  a  Latin  word,  meaning  more. 


12  ADDITION.  [CHAr.  HI. 

Let  the  pupil  commit  to  memory  the  following 


ADDITION  TABLE. 


2+  2=  4 

8- 

-  2=  5 

4+  2=  6 

5+  2=  7 

C+   2=  S 

2-j-  3=  5 

;V 

-   3=  6 

4+  3=  7 

5+  3=  8 

6+  3=  9 

24-  4=  6 

3- 

-  4=  7 

4+  4=  8 

5+  4=  9 

6+  4=10 

2+  5=  7 

3- 

-  5=  8 

4+  5=  9 

5+  5=10 

6+  5=11 

2+  6=  8 

Jj+   6=  9 

4+  6=10 

5+  6=11 

6+  6=12 

2+  7=  9 

3+  7=10 

4+  7=11 

5+  7=12 

6+  7=13 

2+  8=10 

3- 

-  8=11 

4+  8=12 

5+  8=13 

6+  8=14 

2+  9=11 

3- 

-  9=12 

4+  9=13 

5+  9=14 

6+  9=15 

2+10=12 

8- 

-10=13 

4+10=14 

5+10=15 

6+10=16 

2-j-H=13 

3- 

-11=14 

4+11=15 

5+11=16 

6+11=17 

2+12=14 

3+12=15 

4+12=16 

5+12=17 

6+12=18 

7+  2=  9 

8+  2=10 

9+  2=11 

11+  2=13 

12+  2=14 

7+  3=10 

8+  3=11 

9+  3=12 

11+  3=14 

12+  3=15 

7+  4=11 

8+  4=12 

9+  4=13 

11+  4=15 

12+  4=16 

7+  5=12 

8+  5=13 

9+  5=14 

11+  5=16 

12+  5=17 

7+  6—13 

8+  6—14 

9+  6=15 

11+  6=17 

12+  6=18 

74-  7=14 

8+  7=15 

9+  7=16 

11+  7=18 

12+  7=19 

7+   8=15 

8+  8=16 

9+  8=17 

11+  8=19 

12+  8=20 

7+   9=16 

8+  9=17 

9+  9=18 

11+  9=20 

12+  9=21 

7+10=17 

8+10=18 

9+10=19 

11+10=21 

12+10=22 

7+11=18 

8+11=19 

9+11=20 

11+11—22 

12+11=23 

7+12=19 

8+12=20 

9+12=21 

11+12=23 

12+12=24 

i 

. — Let  the  foregoing  table  be  thoroughly  committed  to  mem- 
ory. Time  bestowed  here  will  save  time  in  all  after  arithmetical 
processes.  The  preceding  combinations  of  digits  lie  at  the  founda- 
tion of  the  combinations  of  larger  numbers.  Thus,  if  the  scholar  see 
instantly  that  7  and  6  are  equal  to  13,  he  will  also  instantly  see  that 
17  and  6  are  equal  to  23 ;  the  right-hand  figure  being  the  same  in 
all  sums  that  differ  by  one  or  more  tens. 

It  will  be  well  for  the  teacher  to  skip  about  in  examining  the  pu- 
pil, and  to  require  prompt  answers,  preventing,  if  possible,  the  ob- 
taining of  the  sum  required  by  adding  one  at  a  time. 

After  having  exercised  the  pupil  in  adding  two  numbers,  let  the 
question  combine  three,  four,  or  more  ;  thus,  7+8+6+9  ? 


§11.] 


ADDITION. 


IS 


§11.  Where  the  sums  of  the  several  columns  are  less 
than  ten  :  — 

Add  together  2432,  3343,  and  4122.. 

Write  the  numbers  so  that  the  figures  of  the  same  kind  shall  fall 
in  the  same  column  ;  that  is,  so  that  units  may  be  under  units,  tens 
under  tens,  <fcc.  Draw  a  line  under  the  whole,  as  in  the  example. 


Add,  first,  the  column  of  units.  "Write  the 
sum,  7,  underneath.  Next,  add  the  tens  ;  write 
their  sum,  9,  under  the  column  of  tens.  Next, 
add  the  hundreds  ;  write  the  sum,  8,  under  the 
hundreds.  Lastly,  add  the  thousands,  and  set  the 
sum,  9,  under  the  column  of  thousands.  -The 
whole  sum  is  nine  thousand  eight  hundred  and 
ninety-seven. 


1.  Add  6264,  2532,  and  1203. 

2.  Add  4132,  1001,  and  1423. 
Add  the  following  : 

(3.)       (4.)       (5.) 

5153      3642      2001 
4245      6037      3658 


3  I  «  w 

os:*; 

?«  5  r»  _« 
H  B  H  & 

2432 
3343 
4122 

9897 


3026178 
2513421 


NOTE. — Let  the  pupil  prepare  himself  in  his  seat  to  perform  the 
addition  in  the  class.  At  the  recitation,  let  one  be  required  to  add, 
say,  the  sums  in  Ex.  3.  The  scholar  called  upon,  will  answer,  at 
once,  eight,  nine,  three,  nine,  which  figures  the  class  will  write  in 
their  proper  places. 


§  12.  Where  the  sums  of  the  several  columns  equal  or 
exceed  ten : — 

Find  the  sum  of  the  following  numbers  :  3758,  4903, 
7006,  3713,  3721. 

-3  • 


23101 


14:  ADDITION.  [CHA.P.  IIL 

Place  the  numbers  as  directed  in  (§11)  the 
preceding  example.  The  sum  of  the  numbers 
in  the  units'  column  is  21 ;  that  is,  2  tens  and  1 
unit.  Set  the  1  under  the  units'  column,  and 
carry  the  2  to  the  next,  or  tens'  column.  The 
sum  of  the  tens'  column,  thus  increased,  is  10 
tens ;  that  is,  1  hundred  and  no  tens.  Place  a 
naught  under  the  tens'  column,  and  carry  the  1 
to  the  hundreds'  column.  The  sum  of  the  hun- 
dreds'column,  so  increased, is  31  hundreds;  that 
is,  3  thousands  and  1  hundred.  Set  the  1  under 
the  hundreds'  column,  and  carry  the  3  to  the 
thousands'  column.  The  sum  of  this  column,  so  increased,  is  23 
thousands,  or  2  tens  of  thousands  and  3  thousands.  Set  the  3 
under  the  thousands'  column,  and  carry  the  2  to  the  tens  of  thou- 
sands' place ;  or,  what  is  the  same  thing,  set  down  the  whole  of  the 
sum  of  the  last  column. 

From  what  has  thus  far  been  explained,  the  pupil  will  be 
able  to  understand  the  following 

RULE. 

I.  Place  the  numbers  to  be  added  so  that  the  figures  of  the 
same  kind  shall  fall  in  the  same  column. 

II.  Commence   at  the  right,  and  add  each  column  suc- 
cessively:  if  the  sum  of  any  column  be  less  than  10,  place 
each  sum  under  the  column  added;  but  if  it  equal  or  exceed 
10,  place  the  right-hand  figure  of  the  sum  under  the  column 
added,  and  carry  the  left-hand  figure  or  figures  to  the  next 
column. 

III.  Write  down  the  whole  sum  of  the  last  column. 

PROOF    OF    ADDITION. 

Begin  at  the  top,  and  add  the  columns  downward.  This 
will  vary  the  order  in  which  the  figures  were  added,  and 
will  be  likely  to  rectify  error. 


§12.] 


ADDITION. 


15 


EXAMPLES. 


Prepare  the  following  before  recitation,  and  let  them  be 
performed  aloud  in  the  class  ;  thus  (Ex.  8),  "  twelve,  twen- 
ty :"  the  class  writes  the  proper  figure  in  its  place  ;  "  four, 
nine,  eighteen,  twenty-one,"  and  so  on. 


(8.) 

(9.) 

56430 

7921341 

12798 

82345678 

34457 

79013265 

21325 

7890275 

(10.) 

62849765 
98765432 
23456789 
64698705 


(11.)  - 

2809576832 

95300765 

298 

46543298764 


(12.) 

7920658437 
5987693219 
1439758436 
2870129876 
5432194562 


(13.) 

49827088221 

30765098762 

25421765765 

379866 

2996 


(14.) 

54952138017 
79876320441 
57632768719 
28495762804 
998799 


(15.) 

34567890 

2357911 

234567 

24897 

64 


(16.) 

43345678 

21123355 

27893 

54689 

734321 


(17.) 

123423434 

23785432 

9876543 

751002 

10200 


18.  Add  123405,  2354210,  794327,  and  36547,  together 

19.  Add  275602,  345607,  4567801,  and  365,  together. 

20.  Add  100375,  406780,  4673005,  4112,  and  2478,  to- 
gether. 


16  ADDITION.  [CHAP.  III. 

21.  Add  1034001,  78954,  379205,  367001,  and  45637, 
together. 

22.  What  is  the  sum  of  the  following  numbers :  Three 
thousand  six  hundred  and  fifty,  seven  thousand  eighl,  hun- 
dred and  thirty-two,  eleven  thousand  five  hundred  and  sixty- 
seven,  ten  thousand  and  fifty-six,  four  hundred  and  seventy- 
two  ? 

23.  What  is  the  sum  of  the  numbers,  four  thousand  three 
hundred  and  seventy-three,  three  thousand  one  hundred  and 
fourteen,  one  thousand  two  hundred  and  twenty-three,  six 
hundred  and  fifty-four  ? 

24.  Find  the  number  of  days  in  a  year — the  days  of  the 
respective  months  being  as  follows:  January  31,  February 
28,  March  31,  April  30,  May  31,  June   30,   July  31,  Au- 
gust 31,  September  30,  October  31,  November  30,  Decem- 
ber 31. 

25.  A  man  drew  five  loads  of  bricks  :  in  the  first  load 
he  had  1209,  in  the  second  load   1453,  in  the  third  load 
1101,  in  the  fourth  load  1212,  and  in  the  fifth  load  1303. 
How  many  bricks  were  there  in  all  ? 

26.  If  there  are  shipped  from  the  United  States,  15624 
barrels  of  flour  to  Sweden,  250  barrels  to  Holland,  205154 
barrels  to  England,  6401  to  Texas,  19602  to  Mexico,  what 
is  the  whole  amount  ? 

27.  In  1837  the  United  States  exported   100232  hogs- 
heads of  tobacco;  in  1838  they  exported  100592  ;  in  1839 
they  exported  78995  ;  in  1840  they  exported  119484  ;  in 
1841  they  exported  147828.     How  many  hogsheads  of  to- 
bacco were  exported  during  these  five  years  ? 

28.  If  the  cotton  crop  of  the  United  States  is  estimated 
at  1360532  bales  for  the  year  1839,  2177835  bales  for  the 
year  1840,  1634945  bales  for  the  year  1841,  and  1683574 


§  12.]  ADDITION.  17 

bales  for  the  year  1842,  how  many  bales  will  the  four  years' 
crops  amount  to  ? 

29.  In  1839  the  Onondago  Springs  produced  2864718 
bushels  of  salt;  in  1840  they  produced  2622305  bushels  ; 
"in  1841   they  produced    3340769    bushels;    in  1842  they 
produced  2291903  bushels.     What  is  the  whole  number  of 
bushels  during  the  above  four  years  ? 

30.  The  United  States  exported  in  bullion  and  specie — in 
1838,  3508046  dollars  ;  in  1839,  8776743  dollars  ;  in  1840, 
8417014  dollars;  in  1841,  10034332  dollars.     How  much 
was  exported  during  these  four  years  ? 

31.  The  amount  of  tea  consumed  in  the  United  States, 
during  1842,  was  1348265  pounds;  in  1843,  it  was  12785748 
pounds  ;  in  1844,  it  was  13054327  pounds  ;  in  1845,  it  was 
17162550  pounds;  and  in  1846,  it  was  16891020  pounds. 
What  was  the  whole  number  of  pounds  during  these  five 
years  ? 

32.  The  amount  of  coffee  consumed  in  the  United  States, 
during  the  year  1842,  was  107383567  pounds  ;  in  1843,  it 
was  85916666  pounds;  in  1844,  it  was  149711820  pounds; 
in   1845,  it  was   94358939  pounds  ;  and  in   1846,  it  was 
124336054  pounds.    What  was  the  whole  number  of  pounds 
during  these  five  years  ? 

33.  The  number  of  acres  of  public  land  sold  by  the 
United  States  government,  in  the  year  1841,  was  1164796 
acres;  in  the  year  1842,  it  was  1129217  acres;  in  1843,  it 
was  1605264  acres  ;  in  1844,  it  was  1754763  acres  ;  and  in 
1845,  it  was  1843527  acres.     What  was  the  whole  number 
of  acres  sold  during  these  five  years  ? 

34.  The  United  States  revenue  for  letter  postage,  under 
the  new  law,  was  as  follows:  for  the  year  1842,  it  was 
3953315  dollars;    for  1843,  it  was   3738307  dollars;  for 
184-f,  it  was  3676162  dollars  ;  and  for  1845,  it  was  3660231 


18  ADDITION.  [CHAP.  ILL 

dollars.    What  was  the  whole  number  of  dollars  during  these 
four  years  ? 

35.  In  1843,  the  amount  of  gold  coined  at  the  United 
States  mint  and  branches  was  as  follows :  at  Philadelphia, 
4062010  dollars;  at  the  branch  at  New  Orleans,  3177000 
dollars  ;  at  Dahlonega,  582782  dollars;  at  Charlotte,  287005 
dollars.     How  many  dollars  of  gold  were  coined  in  all  ? 

36.  The  population  of  Europe  is  estimated  at  two  hun- 
dred and  thirty-three  millions,  two  hundred  and  forty  thou- 
sand and  forty-three ;  of  Asia,  at  six  hundred  and  eight 
millions,  five  hundred  and  sixteen  thousand  and  nineteen ; 
of  Africa,  at  one  hundred  and  one  millions,  four  hundred 
and   ninety-eight  thousand,  four  hundred  and  eleven ;    of 
America,  at  forty-eight  millions,  seven  thousand  one  hun- 
dred and  fifty  ;  of  Oceanica,  one  million,  eight  hundred  and 
thirty-four  thousand,  one  hundred  and  ninety-four.     What 
is  the  population  of  the  globe  ? 

37.  The  number  of  square  miles  in  Europe  is  estimated 
at  three  millions,  eight  hundred  and  seven  thousand,  one 
hundred  and  ninety-five ;  in  Asia,  seventeen  millions,  eight 
hundred  and  five  thousand,  one  hundred  and  forty-six ;  in 
Africa,  eleven  millions,  six  hundred  and  forty-seven  thou- 
sand, four  hundred  and  twenty-eight ;  in  America,  thirteen 
millions,  five  hundred  and  forty-two  thousand,  four  hundred ; 
in  Oceanica,  three  millions,  three  hundred  and  forty-seven 
thousand,  eight  hundred  and  forty.     What  is  the  number 
of  square  miles  of  land  upon  the  globe  ? 

38.  An  army  consists  of  52714  infantry,  5110  horse,  6250 
dragoons,  3927  light-horse,  928  gunners,  1410  pioneers,  250 
sappers,  and  406  miners.  What  is  the  whole  number  of  men  ? 

The  quantity  and  value  of  teas  and  coffee  consumed  an- 
nually, from  1821  to  1846,  in  the  United  States,  were  as 
follow : 


§  13.] 


ADDITION. 


19 


TEAS  CONSUMED. 

COFFEK  CONSUMED. 

YEARS. 

Pounds. 

Value. 

Pounds. 

Value. 

1821 

4586223 

$1080264 

11886063 

$2402311 

1822 

5305588 

1160579 

18515271 

3899042 

1823 

0474934 

1547695 

16437045 

2835420 

1824 

7771619 

2224203 

20.797069 

2513950 

1825 

7173740 

2346794 

20678062 

1995892 

1826 

8482483 

3443587 

25734784 

2710536 

1827 

3070885 

942439 

28354197 

1130607 

1828 

6289581 

1771993 

39156733 

3695241 

1829 

5602795 

1531460 

33049695 

3052020 

1830 

6873091 

1532211 

38362687 

3180479 

1831 

4656681 

1057528 

75700757 

5796139 

1832 

8627144 

2081339 

36471241 

2516120 

1833 

12927043 

4775081 

75057906 

7525610 

1834 

13193553 

5422275 

44346505 

4473937 

1835 

12331638 

3594293 

91753002 

9381689 

1836 

14484784 

4472342 

77647300 

7667877 

1837 

14465722 

5003401 

76044071 

7335506 

1838 

11978744 

2559546 

82872633 

7138010 

1839 

7748028 

1781824 

99872517 

9006685 

1840 

16860784 

4059545 

86297761 

7615824 

1841 

10772087 

3075332 

109200247 

9855273 

1842 

13482645 

3567745 

107383567 

8447851 

1843 

12785748 

3405627 

85916666 

5923927 

1844 

13054327 

3152225 

149711820 

9054298 

1845 

17162550 

4809611 

94358939 

5380532 

1846 

16891020 

3983337 

124336054 

7802894 

TOTALS. 

39-42.  What  number  of  pounds  of  tea  was  consumed  for 
the  10  years  commencing  1821?  for  the  10  years  com- 
mencing 1831  ?  from  1841  to  1846  inclusive?  What  num- 
ber from  1821  to  1846  inclusive  ? 


20  SUBTRACTION.  [CHAP.  IV. 

43-46.  What  was  the  value  of  teas  consumed  during  the 
same  periods  of  time  ? 

47-50.  What  number  of  pounds  of  coffee  was  consumed 
for  the  10  years  commencing  182]  ?  for  the  10  years  com- 
mencing 1831  ?  from  1841  to  1846  inclusive?  from  1821 
to  1846  inclusive? 

51-54.  What  was  the  value  of  the  coffee  consumed  dur- 
ing the  same  periods  of 'time  ? 

NOTE. — The  preceding  table  furnishes  ample  materials  for  all  ne- 
cessary discipline  of  the  pupil  in  addition.  It  is  recommended  that 
exercises  from  the  table  be  performed  in  the  following  manner.  Let 
one  pupil  be  called  upon  to  add  eight,  ten,  or  twelve  of  the  num- 
bers ;  another  pupil  a  different  set  of  numbers ;  or  ono  pupil  one 
column  of  figures,  another  the  next  column,  and  so  on ;  or  *.et  one 
pupil  add  a  column  upward,  another  add  the  same  column  down- 
ward. Let  these  exercises  be  performed  aloud  :  the  pupil  omitting 
all  intermediate  words,  and  uttering  only  the  sum  as  it  is  increased 
by  the  addition  of  the  successive  figures.  The  pupil  must  not  be  al- 
lowed to  delay  at  each  figure.  If  he  cannot  run  the  column  rapidly, 
he  should  be  drilled  again  in  the  Table,  §  10. 

55-58.  Add  the  columns  in  the  preceding  table,  two  fig- 
ures at  a  time. 

Thus  (commencing  at  the  right  hand),  "  six,  twenty-one, 
twenty-five,  thirty-four,"  &c. 


CHAPTER    IV. 

SUBTRACTION. 

§  13.  SUBTRACTION  is  the  process  of  taking  a  less  number 
from  a  greater,  to  find  the  difference  ;  as,  5  from  14  leaves  9, 

The  greater  number  is  called  the  Minuend  ;  the  smaller 
number  is  called  the  Subtrahend. 

NOTE. — The  termination nd  means  to  be.     Thus  Minuend  means  the 


$  14.]  SUBTRACTION.  21 

number  to  'be  diminished  ;  Subtrahend  means  the  number  to  be  sub- 
tracted. 

The  difference  of  two  numbers  is  also  called  the  remainder. 

The  symbol  for  subtraction  is  — .  Thus  5  — 3  =  2,  is  the 
same  as  if  written  5  diminished  by  3  equals  2.  The  ex- 
pression may  be  read,  five  less*  three  is  two. 

1-12.  How  many  are  3  from  5  ?  3  from  12  ?  3  from  7  ? 
3  from  9  ?  3  from  13  ?  3  from  20  ?  3  from  8  ?  3  from  11  ? 
3  from  10  ?  3  from  14  ?  3  from  19  ?  3  from  17  ? 

13-25.  How  many  are  4  from  6  ?  from  8  ?  10?  20? 
18?  17?  14?  16?  19?  11?  7?  9?  12? 

26-39.  How  many  are  5  from  7?  from  10?  13?  16? 
19?  8?  11?  14?  17?  20?  9?  12?  15?  18? 

40-52.  How  many  are  6  from  8?  from  11?  14?  17? 
20?  9?  12?  15?  18?  10?  13?  16?  19? 

53-64.  How  many  are  7  from  9?  from  12?  15?  18? 
10?  13?  16?  19?  11?  14?  17?  20? 

65-75.  How  many  are  8  from  10?  from  13?  16?  19? 
11  ?  14?  17?  20?  12?  15?  18? 

76-85.  How  many  are  9  from  11?  14?  17?  20?  12? 
15?  18?  13?  10?  19? 

NOTE. — Let  the  pupil  be  exercised  on  the  preceding  questions  un- 
til he  can  answer  with  great  promptness. 

§  14.  In  which  no  figure  of  the  subtrahend  is  larger  than 
the  corresponding  figure  of  the  minuend. 

From  796  subtract  375. 

Place  the  subtrahend  under  the  minu- 
end, as  in  the  example  ;  units  under  units, 


Commence  at  the  units'  column  and  sub- 
tract :  5  from  6  leaves  1  ;  place  the  1  un- 


derneath,  and  so  proceed  with  each  suc- 

421  Difference. 
ceedmg  column. 


796  Minuend. 
375  Subtrahend. 


*  The  symbol  —  is  often  read  minus,  a  Latin  word,  meaning  lest. 


8053 
4967 


3086 


22  SUBTRACTION.  [CHAP.  IV 

(86.)     (87.)     (88.)      (89.)       (90.) 
687     7649     69985     879465     987654321 
486     5438     59831     729355     821350011 

NOTE. — Let  the  pupil  prepare  himself  in  his  seat  to  perform  the 
subtraction  in  the  class.  At  the  recitation,  let  one  be  required  to 
subtract,  as  in  Ex.  88.  The  scholar  called  upon  will  answer  at  once 
the  results,  without  any  intermediate  words,  thus :  "  four,  five,  one," 
<fcc.,  which  figures  the  class  will  write  in  their  proper  places. 

§  15.  In  which  figures  of  the  subtrahend  are  larger  than 
their  corresponding  figures  of  the  minuend. 

From  8053  subtract  4967. 

"We  cannot  subtract  7  from  3 ;  therefore  we  add  10  to 
the  3,  and  say,  7  from  13  leaves  6.  Having  thus  increased 
the  minuend  figure  3  by  10  units,  we  balance  that  excess 
by  adding  1  ten  to  the  6  of  the  subtrahend,  making  7 
tens.  But  the  7  cannot  be  subtracted  from  the  5  tens.  Add,  then, 
10  tens  to  the  5,  making  15  tens,  and  then  say,  7  from  15  leaves  8. 
Having  added  10  tens  to  the  5  of  the  minuend,  we  restore  the  bal- 
ance by  adding  1  hundred  to  the  9  of  the  subtrahend,  making  10. 
But  we  cannot  subtract  10  from  0.  Then  we  add  10  hundred  to  the 
0,  and  say  10  from  10  leaves  0.  Before  subtracting  the  thousands, 
we  must  add  1  to  the  4  thousands,  to  compensate  for  the  10  hun- 
dred added  to  the  0.  We  then  say,  5  from  8  leaves  3.* 

*  The  following  is  another  mode  of  performing  this  example: 

We  cannot  subtract  the  7  from  the  3.  We  there- 
fore take  1  ten  from  the  tens'  figure  of  the  minuend, 
leaving  that  figure,  4  (which  we  place  in  brackets 
over  the  5,  marking  out  the  5),  and  counting  the  1  ^ 

ten  as  ten  units,  we  add  it  to  the  3  units,  making  13 
units,  which  sum  we  place  in  brackets  over  the  3, 
and  mark  out  the  3.  We  can  now  subtract  the 
7  from  the  13.  We  next  seek  to  subtract  the  6 
from  the  4,  which  we  cannot  do.  We  must  then 
seek  one  from  the  hundreds'  place  to  be  added  to 
the  4.  But  there  are  no  hundreds  there.  We  then 
go  to  the  thousands'  place.  Taking  one  from  the  8, 
we  have  7  left.  Place  the  7  in  brackets  over  the  8 
and  mark  out  the  8.  The  1  thousand  we  carry  to 


:     W    [14]    : 

[7]  [10]   [fl   [13] 

0       0         £      3 

4967 


3086 


15.]  SUBTRACTION.  23 


(91.) 

(92.) 

(93.) 

(94.) 

(95.) 

9034 

8087 

87315 

64281 

5987650 

7941 

4759 

19848 

38796 

4898562 

(96.) 

(97.) 

(98.) 

(99.) 

34678 

789347 

10345678937 

9345678201. 

13787 

120305 

902134124 

3279609167 

NOTE. — See  note,  §  14.  Let  the  teacher,  also,  write  exercises  upon 
the  blackboard,  to  be  performed  by  the  pupil,  or  by  the  class  in  uni- 
son. Let  promptness,  as  well  as  accuracy,  be  aimed  at.  While  one 
pupil  is  calling  out  the  results,  another  may  be  appointed  to  watch, 
and  correct  any  error,  while  a  third  may  write  down  the  answers 
with  the  corrections  under  them,  <fcc. 

From  what  has  thus  far  been  explained  in  subtraction,  the 
pupil  will  be  able  to  understand  the  following 

RULE. 

I.  Place  the  less  number  under  the  greater,  so  that  units 
may  stand  under  units,  tens  under  tens,  &c. 

II.  Commence  at  the  right,  and  subtract  each  figure  of  the 
subtrahend  from  the  corresponding  figure  of  the  minuend. 

the  hundreds'  place,  where  it  counts  10  hundred  ;  place  the  10  over  the  naught, 
and  mark  out  the  0.  Then  take  1  hundred  from  the  10  in  the  brackets,  leaving  9, 
which  place  in  second  brackets  above,  and  mark  Out  the  10  ;  then  add  the  1  hun- 
dred, counting  it  as  10  tens,  to  the  4,  and  you  have  14  tens,  which  place  within 
second  brackets  over  the  4  and  mark  out  the  4. 

Now  we  proceed  with  the  subtraction :  6  from  14  leaves  8  ;  9  from  9  leaves  0 ; 
4  from  7  leaves  3. 

It  will  be  noticed  that  the  minuend  appears  in  three  different  forms;  yet  the  sum 
is  the  same  in  all.  Thus,  in  the  minuend  proper,  the  sum  is  8  thousands,  0  hun- 
dreds, 5  tens,  3  units;  in  the  minuend  in  the  first  brackets,  the  sura  is  7  thousands, 
10  hundreds,  4  tens,  13  units ;  in  the  second  brackets,  7  thousands,  9  hundreds,  14 
tens,  13  units ;  each  form  being  equal  to  8053. 

These  explanations  are  intended  to  show  the  reasons  of  the  process.  The  pupU 
Khould  perform  similar  operations  without  writing  down  the  steps. 


24:     ,  SUBTRACTION.  [CHAP.  IV. 

If  a  figure  of  the  subtrahend  be  greater  than  the  correspond- 
ing figure  of  the  minuend,  add  10  to  the  minuend  figure  be- 
fore subtracting,  and  then  carry  1  to  the  next  figure  of  the 
subtrahend. 

PROOF    OF    SUBTRACTION. 

§  16.  If  the  work  be  correct,  the  difference  added  to  the 
subtrahend  will  equal  the  minuend. 

Perform  and  prove  the  first  8  of  the  following 

EXAMPLES. 

(100.)       (101.)       (102.)        (103.) 
7654321    549876509    7950042689    8877665938 
0578906    498763218    6592719897    1988329876 


(104.)       (105.)       (106.)        (107.) 
8947629    976807136    8765421987    4987670981 
98421      7690234     400000901     768432108 


108.  From  seven  millions,  three  hundred  and  sixty-five 
thousand,  two  hundred  and  thirty-nine,  take  three  hundred 
and  forty-two  thousand  and  thirteen. 

109.  From  one  million  and  eleven,  subtract  thirteen. 

110.  From  three  hundred  and  sixty-five  thousand,  take 
three  hundred  and  sixty-five. 

111.  America  was  discovered  in  1492.     How  many  years 
from  that  time  till  the  year  1844. 

112.  If  a  man  receive  11345  dollars,  and  pay  out  of  it 
9203  dollars,  how  much  will  he  have  remaining  ? 

113.  In  1842  the  Onondaga  Salt  Springs  yielded  2291903 
bushels  of  salt,  and  in  1826  they  yielded  827505  bushels. 
How  many  more  bushels  were  produced  in  1842  than  in 
1826? 


§  16.]  SUBTRACTION.  25 

114.  In  1842  the   United    States   shipped   to   England 
205154  barrels  of  flour,  to  Scotland  3830  barrels.     How 
many  more  barrels  were  sent  to  England  than  to  Scot- 
land? 

115.  Two  men  start  together  from  the  same  place,  and 
travel  in  the  same  direction ;  one  goes  63  miles  each  day, 
and  the  other  goes  37  miles.     How  far  apart  will  they  be 
at  the  end  of  the  first  day  ? 

116.  George  Washington  was  born  in  the  year  1732  ;  he 
died  in  the  year  1799.     To  what  age  did  he  live  ? 

117.  At  an  election  12572  votes  are  taken,  of  which  the 
successful  candidate  received  7391.     How  many  votes  did 
the  other  candidate  receive  ? 

118.  And  what  was  the  first  one's  majority? 

119.  The  coinage  of  the  United  States  mint  for  1843  was 
in  value  11967830  dollars,  and  in  1846  it  was  6633965  dol- 
lars.    How  much  greater  in  value  was  the  coinage  in  1843 
than  in  1846  ? 

120.  The  total  number  of  pieces  coined   in    1843    was 
114640582,  and  in  1844  it  was  9051834.     How  many  more 
pieces  were  coined  in  1843  than  in  1844. 

121.  In  the  year  1846,  the  value  of  the  gold  coin  pro- 
duced at  the  mint  was  4034177  dollars  ;  the  value  of  the  sil- 
ver coin  was  2558580    dollars ;  and  the  copper  coin  was 
41208  dollars.     How  much  greater  was  the  value  of  the 
gold  than  the  silver,  and  how  much  greater  than  the  copper  ? 
Also,  how  much  did  the  silver  exceed  the  copper  ? 

122.  In  1835,  the  number  of  post-offices  in  the  United 
States  was  10770;  extent  of  post-roads  112774  miles:  in 
1845,  the  number  of  offices  was  14183  ;  and  extent  of  roads 
143940  miles.     How  many  offices  were  added  during  these 
10   years,   and   how   many  additional  miles  of  road  were 
added  ? 


26  SUBTRACTION.  [CHAP.  IV. 

123.  In  1840  the  population  of  New  York  was  2428921, 
and  in  1830  it  was  1913006.  What  was  the  increase  du- 
ring this  10  years  ? 


QUESTIONS  INVOLVING  ADDITION  AND  SUBTRACTION. 

124.  A.  lets  B.  have  60  bushels  of  wheat,  worth  70  dol- 
lars, a  fine  horse  worth  150  dollars,  and  27  dollars'  worth 
of  butter.     B.  in  turn  gives  A.  his  note  for  110  dollars,  and 
the  rest  in  cash.     What  is  the  amount  of  cash  ? 

125.  A.  borrows  of  B.,  at  one  time,  375  dollars ;  at  a 
second  time  he  borrows  95  dollars,  and  at  a  third  time  he 
borrows  413  dollars ;  he  has  paid  him  319  dollars.     How 
much  does  he  still  owe  him  ? 

126.  A  person  left  a  fortune  of  10573  dollars  to  be  di- 
vided between  two  sons  and  one  daughter ;  the  first  son 
received    4309    dollars,  the   other  son    had  4987  dollars. 
How  much  did  the  daughter  receive  ? 

127.  Two  persons  are  375  miles  apart;  they  travel  to- 
wards each  other  ;  at  the  end  of  one  day,  one  has  travelled 
93  miles  and  the  other  57  miles.     How  far  apart  are  they? 

128.  A  farmer  sold  a  span  of  horses  for  150   dollars,  a 
cow  for  27  dollars,  some  cheese  for  83  dollars,  and  7  tons 
of  hay  for  56  dollars.     He  purchased  10  yards  of  broad- 
cloth worth  45  dollars,  a  cook-stove  for  23  dollars,  and  a 
pleasure  carriage  for  80  dollars.     How  much  money  will 
he  have  left  ? 

129.  In  the  year  1840,  the  coinage  of  the  United  States 
mint  was  as  follows:    1675302  dollars  of  gold,  J 72 6 703 
dollars  of  silver,  and  24627  dollars  of  copper ;  in  the  year 
1841   the  gold  coin  amounted  to  1091597,   the  silver  to 
1132750,  and  the  copper  to  15973.     How  much  was  the 
whole  value  for  each  year  ?     How  much  greater  was  the 


§  17.]  MULTIPLICATION.  27 

whole  coinage  in  1840  than  in  1841  ?  In  each  year,  how 
much  greater  was  the  value  of  the  silver  than  that  of  the 
gold  and  copper  respectively  ? 

130.  A  person  dying,  left  60000  dollars,  to  be  divided 
among  his  widow,  two  sons,  and  two  daughters,  in  the  fol- 
lowing manner  :  To  each  son  he  gave  10500  dollars,  to  each 
daughter  9250  dollars,  and  the  residue  to  the  widow.  What 
was  the  widow's  portion  ? 

131.  In  1836  the  number  of  volumes  in  the  Royal  Li- 
brary of  Paris  was  700000,  and  the  number  of  manuscripts 
80000.     In  the  same  year  the  Vienna  Library  consisted  of 
300000  volumes  and  16000  manuscripts.     How  many  vol- 
umes and  how  many  manuscripts  in  both  libraries  ?     How 
many  more  volumes  and  how  many  more  manuscripts  in  the 
Paris  library  than  in  the  Vienna  ?     The  total  number  of  vol- 
umes exceed  the  total  number  of  manuscripts  by  how  many  ? 

132.  At  a  certain  election  the  number  of  votes  polled 
for  two  opposing  candidates  was  544  and  431  respectively. 
What  was  the  total  number  of  votes  polled,  and  what  was 
the  majority  of  the  successful  candidate  ? 


CHAPTER    V. 

MULTIPLICATION. 

§  17.  MULTIPLICATION  is 'the  process  of  repeating  one  of 
two  numbers  as  many  times  as  there  are  units  in  the  other. 

The  number  to  be  repeated  is  called  the  multiplicand. 

The  number  denoting  how  many  times  the  multiplicand 
is  to  be  repeated  is  called  the  multiplier. 


MULTIPLICATION. 


[CHAP,  v 


Both  multiplicand  and  multiplier  are  called  factor**  The 
result  obtained  is  called  the  product. 

The  symbol  for  multiplication  is  X .  Thus  3x"7  means 
3  times  7,  or  3  multiplied  by  7. 


MULTIPLICATION  TABLE. 


2X  1=  2 

4X  1=  4 

6X  1=  6 

8X  1=    8 

nx  1=  n 

2X  2=  4 

4X  2=  8 

6X  2=12 

8X  2=  16 

JlX  2=  22 

2X  3=  6 

4X  3=12 

6X  3=18 

8X  3=  24 

11  X  3=  33 

2X  4=  8 

4X  4=16 

6X  4=24 

8X  4=  32 

11  X  4=  44 

2X  5=10 

4X  5=20 

6X  5=30 

8X  5=  40 

11  X  5=  55 

2X  6=12 

4x  6=24 

6x  6=36 

8X  6=  48 

11  X  6=  66 

2X  7=14 

4X  7=28 

6X  7=42 

8x  7=  56 

11X  7=  77 

2X  8=16 

4x  8=32 

6X  8=48 

8X  8=  64 

11  X  8=  88 

2X  9=18 

4X  9=36 

6X  9=54 

8X  9=  72 

11X  9=  99 

2X10=20 

4x10=40 

6X10=60 

8X10=  80 

11X10=110 

2X11=22 

4x11=44 

6X11=66 

8X11=  88 

11X11=121 

2X12=24 

4X12=48 

0X12=72 

8X12=  96 

11X12=132 

3X  1=  3 

5X  1=  5 

7X  1=  7 

9X  1=    9 

12X  1=  12 

3X  2=  6 

5X  2=10 

7X  2=14 

9X  2=  18 

12  X  2=  24 

3X  3=  9 

5X  3=15 

7x  3=21 

9X  3=  27 

2X  3=  36 

3X  4=12 

5X  4=20 

7X  4=28 

9X  4=  36 

2X  4=  48 

3X  5=15 

5X  5=25 

7X  5=35 

9X  5=  45 

2X  5=  60 

3X  6=18 

5X  6=30 

7x  6=42 

9X  6=  54 

12  X  6=  72 

3X  7=21 

5X  7=35 

7x  7=49 

9X  7=  63 

2X  7=  84 

3X  8=24 

5X  8=40 

7X  8=56 

9x  8=  72 

2X  &=  96 

3X  9=27 

5X  9=45 

7X  9=63 

9X  9=  81 

12  X  9=108 

3X10=30 

5X10=50 

7X10=70 

9X10=  90 

12X10=120 

3X11=33 

5X11=55 

7X11=77 

9X11=  99 

12X1  i=l  32 

3X12=36 

5X12=60 

7X12=84 

9X12=108 

12X12=144 

I 

The  product  of  a  factor  multiplied  by  itself  is  called  a  square.  A 
factor,  which,  multiplied  by  itself,  produces  a  given  number,  is  called 
the  square  root  of  that  number.  Thus  16  is  the  square  of  4,  because 
it  is  the  product  of  4  by  itself;  4  is  the  square  root  of  16,  because  4 
multiplied  by  itself  produces  16. 

EXAMPLES. 

1-11.  What  is  the  square  of  2  ?  of  3  ?  of  4  ?  of  5  ?  of 
6  ?  of  7  ?  of  8  ?  of  9  ?  of  10  ?  of  11  ?  of  12  ? 

12-22.  What  is  the  square  root  of  25  ?  of  49  ?  of  4  ? 

*  From  a  Latin  word,  signifying  to  make ;  because  multiplied  together  they 
make  the  product. 


§  18.]  MULTIPLICATION.  29 

of  9  ?  of  16  ?  of  36  ?  of  64  ?  of  81  ?  of  100  ?  of  144  ?  of 
121? 

23-27.  Multiply  8  by  the  square  root  of  16;  9  by  the 
square  root  of  4  ;  12  by  the  square  root  of  49  ;  8  by  the 
square  root  of  64  ;  9  by  the  square  root  of  81. 

28-36.  How  many  are  3  times  4  ?  3  times  6  ?  8  ?  10  ? 
12?  3?  9?  7?  5? 

37-47.  How  many  are  4  times  2  ?  4?6?8?10?12? 
3?  5?  7?  9?  11? 

48-58.  Multiply  by  5  the  following  numbers  :  3  ;  6  ;  9  ; 
12;  2;  5  ;  8;  11  ;  4;  7;  10. 

59-69.  Multiply  by  6  the  following  numbers:  3  ;  6  ;  9  ; 
12;  2;  5;  8;  11;  4;  7  ;  10. 

70-80.  Multiply  by  7  the  following  numbers  :  3  ;  6  ;  9  ; 
12;  2;  5;  8;  11  ;  4;  7;  10. 

8 1-1 3 5.  v  Multiply  the  following  numbers  :  3  ;  6  ;  9  ;  12  ; 
2;5;8;11;4;7;  10— first  by  8  ;  then  by  9  ;  then  by 
10;  then  by  11  ;  then  by  12. 

136-142.  What  numbers  must  be  written  on  the  right 
of  the  sign  =  in  the  following  expressions  ?  6x5=  ; 
6X6=  ;  9x2  —  4=  ;  (6  — 3)*X5  =  ;  (8  +  2)x5  =  ; 
(4  +  9  —  2)xll=  ;  (12  +  4  —  Il+3)x7  =  . 

NOTE. — The  pupil  will  remember  that  the  multiplier  and  the  mul- 
tiplicand may  be  interchanged  without  altering  the  product ;  thus, 
8X7=7X8=56. 

§  18.  When  the  multiplier  consists  of  one  figure. 


*  When  a  parenthesis  is  made  to  embrace  two  or  more  terms,  the  whole  is  to  be 
considered  as  one  quantity.  In  the  above  expression,  the  6  —  3,  which  is  3,  is  to 
be  multiplied  by  5;  so  in  the  expression  (12  +  4—  11  +3)  X  7,  we  first  find  tha 
value  of  12+  4—  11  +  3,  which  is  8,  and  then  multiply  8  by  7,  and  find  56. 

3* 


MULTIPLICATION. 


[CHAP,  v 


Multiply  697  by  3. 

Place  the  multiplier  under  the  unit 
figure  of  the  multiplicand.  First, 
multiplying  the  7  units  by  the  3  units, 
we  obtain  21  units,  or  two  tens  and 
1  unit.  We  write  the  1  unit  under 
the  units'  column,  and  reserve  the  2 
tens  for  the  tens'  column.  We  next 


697  Multiplicand- 
3  Multiplier. 


2091  Product. 


multiply  the  9  tens  l?y  the  3 — the  product  is  27  tens ;  adding  the 
2  reserved,  we  have  29  tens,  or  2  hundreds  and  9  tens.  Write  the  9 
under  the  tens'  columns,  and  reserve  the  2  to  carry  to  the  hundreds 
Finally,  we  multiply  the  6  hundreds  by  the  3,  and  find  the  product 
18  hundreds,  to  which,  adding  the  2  reserved,  we  have  20  hundreds, 
or  2  thousands  and  0  hundreds.  Write  the  naught  under  the  hun- 
dreds and  the  2  in  the  thousands'  place.  Our  product  then  is  2091  * 


(143.)    (144.)     (145.)     (146.)     (147.) 
1234    234156    612378    897654    1003456 
2345         6 

(148.)       (149.) 
205670678   6531023456 

7          8 

(150.)        (151.) 
891030756078   6289376 

9        7 

NOTE. — Let  the  pupil  prepare,  in  his  seat,  to  perform  the  preceding 
examples  in  his  class.  The  recitation  will  consist  in  his  answering 
the  figures  successively,  as  they  are  to  be  written  in  the  product. 
Let  the  teacher  dictate  examples  for  the  slate,  and  make  free  use  of 
the  blackboard  until  the  operations  can  be  performed  very  promptly 
and  accurately. 


*  Annexed  is  a  different  illustration  of  the  principles 
involved  in  the  above  operation.  The  product  of  the  7 
by  the  3  is  21  units,  or  2  tens  and  1  unit ;  the  product  of  the 
9  tens  by  the  3  is  27  tens ;  the  product  of  the  6  hundreds 
by  the  3  is  18  hundreds  :  which  are  written  in  their  appro- 
priate columns  and  added  together.  The  whole  product 
is  as  before. 


697  Multiplicand. 
3  Multiplier. 

21  Units. 

27    Tens. 
18      Hundreds. 

2091  Product. 


§  19.]  MULTIPLICATION.  31 

152-160.  Multiply  31486  by  2;  3;  4;  5;  6;  7;  8; 
9  ;  and  add  the  products, 

161-169.  Multiply  8976201  by  9  ;  8;7;6;5;4;3; 
2  ;  and  add  the  products. 

170-172.  Multiply  652081  by  8,  and  by  3  ;  then  give 
the  difference  of  the  products. 

173-180.  What  is  the  sum  of  the  following  products  ? 
8x7654;  9x872150;  8624  X6  ;  9021763  X9  ;  5765  X3  ; 
1453217x7;  7x7123541. 

§  19.  When  the  multiplier  consists  of  more  than  one 
figure. 

Multiply  367  by  84. 

Place  the  multiplier  under  the  multipli-  ^  Multiplicand, 

cand,  units  under  units,  and  tens  under  tens.  ,,  .  .  .. 

Multiplying  first  by  the  4  units,  we  find 
1468  for  the  product.  We  are  next  to 
multiply  by  the  8  tens.  Now,  it  is  obvi- 
ous that  1  unit,  taken  ten  times,  that  is, 


1468 
2936 

30828  Product. 


multiplied  by  1  ten,  must  produce  10  units 
or  1  ten.  So  7  units  (as  in  the  example), 
multiplied  by  8  tens,  must  produce  56  tens,  or  5  hundreds  and  6 
tens.  Therefore,  set  the  first  figure,  6,  of  this  second  product  under 
the  tens'  column,  and  reserve  the  5  to  carry  to  the  hundreds.  The 
next  step  is  the  multiplication  of  tens  by  tens,  which  must  produce 
hundreds,  since  1  ten,  taken  1  ten  times,  is  equal  to  1  hundred. 
Therefore  8  tens  times  6  tens  are  48  hundreds  ;  to  which  add  the  5 
hundreds  reserved,  and  we  obtain  53  hundreds ;  equal  to  5  thou- 
sands and  3  hundreds.  Place  the  3  under  the  hundreds'  column,  and 
carry  the  5  to  the  next  column,  and  so  proceed  throughout.  The 
sum  of  these  partial  products  will  give  the  total  product,  30828. 

NOTE. — If  the  multiplier  consists  of  three  figures,  its  left-hand  or 
hundreds'  figure,  multiplied  into  the  units  of  the  multiplicand,  will 
give  hundreds  for  the  first  figure  of  the  product,  which  must  of 
course  be  set  down  under  the  hundreds'  column ;  hundreds  and  tens, 


32  MULTIPLICATION.  [CHAP.  V. 

multiplied  together,  will  give  thousands ;  hundreds  and  hundreds, 
multiplied  together,  will  give  ten  thousands,  <fcc. 

If  the  multiplier  consists  of  four  figures,  its  left-hand  or  thousands 
figure  multiplied  into  units,  will  give  thousands  for  the  first  figure  of 
the  product,  which  must  be  set  down  under  the  thousands'  column. 
Thousands  multiplied  into  tens,  gives  tens  of  thousands ;  into  him 
dreds,  gives  hundreds  of  thousands  ;  and  so  on. 

It  would  be  necessary  to  annex  ciphers  to  the  figures  in  these  sev- 
eral products,  to  show  their  true  places,  if  these  places  were  not  de- 
termined by  the  position  of  the  figures  with  relation  to  other  figures, 
whose  places  are  known. 


§  20.  Take  again  the  first  example,  viz.  the 
multiplication  of  697  by  3.  Since  697  is  to  be 
repeated  3  times,  this  may  be  done  by  writing 
the  697  three  times,  and  then  performing  the  ad- 


697 
697 
697 

2091 


dition  as  in  the  example. 

All  questions  in  multiplication  may  be  performed  by  ad- 
dition. Hence,  multiplication  is  sometimes  defined  as  being 
a  concise  way  of  performing  several  additions. 

§  21.  The  pupil  will  now  be  able  to  apply  understand- 
ingly  the  following 

RULE. 

4 

I.  Place  the  multiplier  under  the  multiplicand,  units  un- 
der units,  &c. 

II.  If  the  multiplier  consists  of  one  figure,  multiply  by  it 
each  figure  of  the  multiplicand  successively.     Place  under- 
neath, the  right-hand  figure  of  each  product,  and  carry  the 
left-hand  figure  or  figures  to  the  succeeding  product. 

III.  If  the  multiplier  consists  of  more  than  one  figure, 
multiply  by  each  figure  successively.     Place  the  right-hand 
figure  of  each  partial  product  under  the  figure  by  which 


§  22.]  MULTIPLICATION.  33 

you  multiply.     The  sum  of  these  partial  products  will  be 
the  total  product  sought. 

PROOF  OF  MULTIPLICATION. 

§  22.  Interchange  multiplicand  and  multiplier,  and  see  if 
the  same  result  "be  obtained. 

Or,  carefully  repeat  the  multiplication. 

EXAMPLES. 


(181.) 

(182.) 

(183.) 

(184.) 

36984176 

809653217 

97654321 

87509313349 

39 

68 

97 

45 

(185.) 

(186.) 

(187.) 

(188.) 

80265947 

79829054 

320008999 

1357902684 

356 

289 

794 

936 

NOTE. — See  note  under  §  18. 

189-195.  Multiply  123456789  by  4287;  by  98321 ;  by 
65943  ;  by  876542  ;  by  7689769  ;  by  987654321  ;  and 
add  their  products. 

196.  What  is  the  difference  of  20946781x87634  and 
598421765X308%? 

197-198.  Write  the  proper  number  on  the  right  of  the 
sign  =  :     (9426176  — 4319832)  X  3426  = 
69428  X  876  —  98140  X  34  = 

199-200.  Multiply  69821  by  the  square  of  22  ;  by  the 
square  root  of  144. 

A 
§  23.  We  know  from  what  has  been  said  (§  6),  that  the 

annexing  of  a  naught  to  a  number  is  the  same  as  multiply- 


34:  MULTIPLICATION.  [CHAP.  V 

ing  that  number  by  10 ;  that  the  annexing  of  two  naughts 
is  the  same  as  multiplying  by  100,  <fec.  Hence, 

When  the  multiplier  or  multiplicand,  or  both,  have  one 
or  more  naughts  at  the  right, 

Multiply  by  the  significant  figures,  as  by  §  21,  and  to  the 
product  annex  as  many  naughts  as  there  are  at  the  righl 
of  both  multiplier  and  multiplicand. 

EXAMPLES. 

201-204.  Multiply  76429  by  10;  by  100;  by  1000; 
by  10000. 

205-208.  Multiply  98740  by  20  ;  by  200  ;  by  2000  ; 
by  20000. 

209-212.  Multiply  654200  by  30  ;  by  300  ;  by  3000  ; 
by  30000. 

213-216.  Multiply  800032000  by  40;  by  400  ;  by  4000; 
by  40000. 

217,  218.  Multiply  307210000  by  3780000;  by 
93600700. 

§  24.  When  the  multiplier  is  a  composite  number. 

A  composite  number  is  one  which  may  be  produced  by 
multiplying  two  or  more  numbers  together.  Thus,  35  is  a 
composite  number,  since  it  may  be  produced  by  multiply- 
ing together  5  and  7.  The  5  and  7  are  called  the  compo- 
nent parts  or  factors  of  35. 

Multiply  48  by  35.  If  we  multiply  48  by  5,  one  of  the 
factors  of  35,  we  obtain  240  for  a  product.  This  product 
multiplied  by  7,  the  other  factor  of  35,  will  give  1680  ;  that 
is,  1680  is  7  times  5  times  48,  or  35  times  48.  Hence  this 

KULE. 
Multiply  the  given  number  by  one  of  the  factors  of  the 


§24.J 


MULTIPLICATION.  35 


multiplier,  and  the  product  thus  obtained  by  another  factor, 
and  so  on.     The  last  product  will  be  the  one  sought. 

EXAMPLES. 

219.  Multiply  365  by  28.  The  factors  of  28  are  4  and 
7,  or  2  and  2  and  7. 

220-224.  Multiply  374  by  24  ;  by  18  ;  by  36  ;  by  63  ; 
by  108. 

225-230.  Multiply  987623  by  84  ;  by  35  ;  by  81  ;  by 
77  ;  by  64  ;  by  40. 

231-237.  Multiply  829  by  144  ;  by  21  ;  by  99  ;  by  30  ; 
by  15  ;  by  24  ;  by  12. 

MISCELLANEOUS    EXAMPLES  I    INVOLVING    ADDITION,    SUBTRAC- 
TION,   AND    MULTIPLICATION. 

238.  Suppose  I  buy  15  loads  of  bricks,  each  load  con- 
taining 1250  bricks  :  how  many  bricks  have  I  ? 

239.  In  an  orchard  there  are  107  apple- trees,  each  pro- 
ducing 1 9  bushels  of  apples.     How  many  bushels  does  the 
whole  orchard  yield  ? 

240.  If  a  person  travel  17  days  at  the  rate  of  37  miles 
each  day,  how  many  miles  will  he  travel  in  all  ? 

241.  If  a  person  buy  175  barrels  of  salt,  each  weighing 
304  pounds,  how  many  pounds  in  all  will  he  have  ? 

242.  Suppose  I  purchase  the  following  bill  of  merchandise : 

3  Firkins  of  butter,  at  15  dollars  each. 
7  Hogsheads  of  molasses,  at  23  dollars  each. 
12  Bags  of  coffee,  at  11  dollars  each. 
5  Boxes  of  raisins,  at  2  dollars  each. 
3  Boxes  of  lemons,  at  5  dollars  each. 

How  many  dollars  must  I  give  for  the  whole  ? 


36  MULTIPLICATION  [CHAP.  V. 

243.  How  many  dollars  will  the  following  bill  of  goods 
amount  to  ? 

52  Yards  of  black  broadcloth,  at  4  dollars  per  yard. 

40  Yards  of  Brussels  carpeting,  at  2  dollars  per  yard 

2  Sofas,  at  56  dollars  each. 

9  Mahogany  chairs,  at  5  dollars  each. 

5  French  bedsteads,  at  7  dollars  each. 

244.  If  the  railroad  extending  between  Albany  and  Buf- 
falo, a  distance  of  326  miles,  cost  25649  dollars  per  mile, 
what  was  the  entire  cost  ? 

245.  How  many  bushels  of  potatoes  may  be  produced 
from  13  acres  of  land,  if  each  acre  produces  212  bushels? 

246.  How  much  must  be  paid  for  constructing  18  miles 
of  plank-road,  at  4211  dollars  per  mile? 

247.  How  much  will  543  cords  of  wood  cost,  at  5  dollars 
per  cord  ? 

248.  In  one  year  there  are  8766  hours.     How  many  hours 
in  1848  years? 

249.  In  one  cubic  foot  there  are  1728  cubic  inches.    How 
many  cubic  inches  in  1 7  cords  of  wood,  each  cord  containing 
128  cubic  feet? 

250.  What  will  13  square  miles  of  land  cost,  at  17  dol- 
lars per  acre,  there  being  640  acres  in  one  mile  ? 

251.  How  many  miles  will  a  steam  locomotive  pass  in  7 
days  of  24  hours  each,  if  it  move  at  the  rate  of  45  miles  each 
hour? 

252.  If  the  earth  move  in  its  orbit  68000  miles  per  hour, 
how  far  will  it  move  in  365  days  of  24  hours  e*ach  ? 

253.  If  one  mile  of  railroad  require  116  tons  of  iron,  worth 
53  dollars  per  ton,  what  will  be  the  cost  of  sufficient  iron  to 
construct  a  road  of  78  miles  in  length? 

254.  In  an  orchard  of  105  apple-trees  the  average  pro- 


§  2i.]  MULTIPLICATION.  37 

duce  of  each  tree  is  7  barrels  of  fruit,  worth  3  dollars  per 
barrel.     What  was  the  income  of  the  orchard  ? 

255.  A  ship,  after  sailing  23  hours  with  the  velocity  of 
8  miles  per  hour,  encounters  a  storm,  by  which  she  is  driven 
directly  back  during  5  hours,  with  a  velocity  of  15  miles  per 
hour.     At  the  end  of  the  28  Sours,  how  far  is  the  ship  from 
the  place  of  departure  ? 

256.  A  man   owing   1213  dollars,  gives  in  payment  3 
horses,  valued  at  150  dollars  each,  15  cows,  at  27  dollars 
each,  and  143  dollars  in  cash.     How  much  remains  unpaid? 

257.  A  person  purchased  a  farm  of  126  acres,  at  36  dol- 
lars per  acre:  at  one  time  he  sold  57  acres,  for  43  dollars 
per  acre  ;  at  another  time  he  sold  37  acres,  for  51  dollars 
per  acre.     How  many  acres  has  he  left,  and  at  what  cost  ? 

258.  A  person  having  a  journey  of  600  miles  to  perform, 
travels  47  miles  daily  for  5  days  ;  during  the  next  4  days 
he  travels  54  miles  each  day.     How  many  more  miles  must 
he  go  to  complete  his  journey  ? 

259.  If  I  give  3  horses,  each  worth  150  dollars,  and  13 
cows,  each  worth  29  dollars,  for  50  acres  of  land,  valued  at 
16  dollars  per  acre,  do  I  gain  or  lose,  and  how  much  ? 

260.  In  one   year  there  are  31557600  seconds.      How 
much  does  one  trillion  exceed  the  number  of  seconds  in 
1850  years? 

261.  A  person  bought  two  farms :  one  of  87  acres,  at 
54  dollars  per  acre,  the  other  of  101  acres,  at  37  dollars  per 
acre  :  he  paid  8140  dollars.     How  much  is  still  due  ? 

262.  How  many  cubic  inches  in  a  hogshead  of  wine  of 
63  gallons,  there  being  231  cubic  inches  in  one  gallon? 

4 


38  DIVISION.  [CHAP.  vi. 


CHAPTER    VI. 

DIVISION. 

§  25.  DIVISION  is  the  process  of  finding  how  many  times 
one  number  is  contained  in  another. 

The  number  to  be  divided  is  called  the  dividend. 

The  number  by  which  we  divide  is  called  the  divisor. 

The  number  of  times  which  the  dividend  contains  the  di- 
visor is  called  the  quotient. 

The  number  which  is  left  over  after  the  division  is  per- 
formed, is  called  the  remainder. 

The  symbol  for  division  is  -f- .  Thus  8  -=-  2,  means  8  di- 
vided by  2.  Division  is  also  often  represented  by  placing 
the  divisor  under  the  dividend  with  a  line  between  them ; 

o 

thus  —  denotes  that  8  is  to  be  divided  by  2. 

2 

NOTE. — This  symbol  is  often  employed  to  express  an  accurate  quo- 
tient where  there  is  a  remainder  after  division.  Thus  13  -f-  3  =4£ ; 
which  means  that  the  quotient  of  13  divided  by  3  is  4  ;  to  which  the 
remainder  1,  divided  by  3,  or  a  third  part  of  1,  is  to  be  added. 

1-12.  How  many  times  is  2  contained  in  2  ?  in  6  ?  in  &? 
in  12  ?  in  4  ?  in  10  ?  in  14  ?  in  18  ?  in  16  ?  in  20  ?  in  24  ? 
in  22? 

13-24.  How  many  times  is  3  contained  in  3  ?  in  9  ?  in 
15  ?  in  21  ?  in  27  ?  in  33  ?  in  36  ?  in  6  ?  in  12  ?  in  18  ? 
hi  24?  in  30? 

25-34.  How  many  times  is  4  contained  in  8  ?  in  16  ?  24  ? 
32?  40?  48?  12?  20?  28?  36? 

35-45.  How  many  times  is  5  contained  in  10  ?  20  ?  30  ? 
40?  50?  60?  15?  25?  35?  45?  55? 


§  26.]  DIVISION.  39 

46-54.  How  many  times  is  6  contained  in  12  ?  24  ?  36  ? 
48?  60?  18?  30?  42?  54? 

55-65.  How  many  times  is  7  contained  in  14?  28?  42? 
56?  70?  84?  21?  35?  49?  63?  77? 

66-76.  How  many  times  is  8  contained  in  16?  32?  48? 
64?  80?  96?  24?  40?  56?  72?  88? 

77-87.  How  many  times  is  9  contained  in  18?  36?  54? 
72?  27?  45?  63?  81?  99?  90?  108? 

88-100.  How  many  times  is  10  contained  in  20  ?  40  ?  60  ? 
90?  80?  70?  30?  50?  100?  130?  120?  110?  140? 

101-109.  How  many  times  is  11  contained  in  22  ?  in  44  ? 
in  66  ?  in  33  ?  in  77  ?  in  110  ?  in  88  ?  in  121  ?  in  132  ? 

110-120.  How  many  times  is  12  contained  in  24?  48? 
60?  96?  84?  144?  120?  132?  36?  72?  108? 

121-129.  What  is  the  quotient  of  108  -r-  12  ?  36  +-  9  ? 

77  +  11?  63  +  9?  100  +  5?  ^?  y?  y?  y? 

130-134.  What  numbers  are  to  be  written  on  the  right 
of  the  symbol  —  in  the  following  expressions?*  (5  +  7)  +  4 
=  ?  (6  +  8  —  2)  +  3  =  ?  (12  +  12  — 
(8  —  4-f-7)x4  +  ll:=  ?  [(9  —  3) 


§  26.  When  the  divisor  consists  of  one  figure. 
Let  us  divide  973  by  7. 


Arrange  the  numbers  as  in  the  example. 
First,  see  how  many  times  the  7  is  contained 
in  the  9  hundreds :  we  find  it  contained  1 
time,  and  2  hundreds  remainder.  "Write  the 
1  underneath.  To  the  next  figure,  7,  which  is 


5     S 
7  )  973 


139  Quotient, 


tens,  add  the  2  hundreds,  or  20  tens,  remain- 

der.    "We  then  have  27  tens,  the  same  result  as  if  we  had  prefixed 

the  2  to  the  7.     Next,  we  see  how  many  times  7  is  contained  in  27 

*  See  note  at  the  bottom  of  page  29. 


4:0 


DIVISION. 


[CHAP.  vi. 


which  is  3  times,  and  6  tens  remainder ;  we  place  the  3  for  the  next 
figure  of  the  quotient,  and  conceive  the  6  to  be  prefixed  to  the  next 
figure  of  the  dividend,  making  63,  which  is  the  same  as  adding  6 
tens  or  60  units  to  the  3  units.  Finally,  we  find  7  is  contained  in  63 
9  times. 

We  find,  then,  7  to  be  contained  in  973,  139  times.  Hence  7  re- 
peated 139  times,  or,  what  is  the  same  thing,  139  repeated  7  times 
must  equal  973. 

NOTE. — The  above  example  might  have  been  performed  another 
way.  We  know  that  7  is  contained  in  973  at  least  once.  Subtract- 
ing 7  from  973,  we  have  966  for  a  remainder.  7  must  be  contained 
in  966  at  least  once.  Subtracting  7  from  966  we  have  959  for  a  re- 
mainder. We  might  go  on  thus  until  we  had  performed  139  sub- 
tractions, which  would  exhaust  the  number,  973.  Hence  7  would 
be  contained  in  973,  139  times.  Division  may  therefore  be  called  a 
concise  way  of  performing  several  subtractions. 

EXAMPLES. 

(135.)  (136.)  (137.)  (138.) 

3)36903       4)6728910       5)8931069       6)31894372 

(139.)  (140.)  (141.) 

7 ) 897654321         8 ) 921076865          9 ) 234567890 

(142.)  (143.)  (144.) 

9 ) 76421068          8 ) 765324816          7 ) 890123456 


(145.) 
6)7890123456 


(146.)' 
5)6789012345 


NOTE. — Let  the  pupil  practice  on  the  preceding  and  similar  exam- 
ples, performing  the  operation  aloud  in  the  class,  by  naming  at  once 
the  successive  quotients  which  go  to  make  the  entire  quotient. 


§  26.]  DIVISION.  41 

147-154.  Divide  347891  by  2  ;  by  3 ;  4;  5;  6;  7; 
8;  9. 

155-162.  Divide  76541307  by  2;  3;  4;  5;  6;  7; 
8;  9. 

163-170.  Divide  897643  by  2  ;  3  ;  4  ;  5  ;  6  ;  7  ;  8  ;  9. 

171-178.  Give  the  quotients  of  7653908 -f- 2;  ^3;  +4; 
+  5;  -7-6;  -r-Y;  -f-8;  -4-9. 

179-183.  Divide  the  following  numbers  by  6:  543218; 
39876541;  79654870002;  46895318;  9998887. 

184-213.  Divide  the  preceding  numbers  by  3 ;  by  4  ;  7  ; 
8;  9;  5. 

The  foregoing  method  is  called  Short  Division.  The 
pupil  will  now  understand  the  following 

RULE. 

I.  Place  the  divisor  at  the  left  of  the  dividend,  keeping 
them  separate  by  a  curved  line,  and  draw  a  straight  line 
underneath  the  dividend. 

•  II.  Seek  how  many  times  the  divisor  is  contained  in  the 
left-hand  figure  or  figures  of  the  dividend,  and  place  the 
result  directly  beneath,  for  the  first  figure  of  the  quotient. 

III.  If  there  is  no  remainder,  divide  the  next  figure  of  the 
dividend  for  the  next  figure  of  the  quotient.  But  when  there 
is  a  remainder,  conceive  it  to  be  prefixed  to  the  next  succeed- 
ing figure  of  the  dividend  before  making  the  next  division. 
If  a  figure  of  the  dividend,  which  is  required  to  be  divided, 
is  less  than  the  divisor,  write  0  in  the  quotient,  and  con- 
sider that  figure  as  a  remainder. 

NOTE. — The  final  remainder,  if  there  be  one,  may  be  placed  over 
the  divisor  with  a  line  between  them,  and  annexed  to  the  quotient. 
(§  25,  note.)  This  will  denote  that  the  remainder  is  still  to  be  di- 
vided by  the  divisor. 

4* 


DIVISION. 


[CHAP.  vi. 


Quotient. 

354  )  4703598  (  13287 
354  1st  product. 

1163  Thousands. 
1062   2d  product. 

1015  Hundreds. 
708    3d  product. 

3079  Tens. 

2832   4th  product. 

2478  Units. 

2478   5th  product. 


§  27.  When  the  divisor  consists  of  more  than  one  figure, 
or  Long  Division. 

Divide  4703598  by  354. 

It  requires  3  figures,  470,  of  the 
dividend,  to  contain  the  divisor. 
This  is  contained  once  in  470 ;  we 
place  the  1  at  the  right  of  the  divi- 
dend for  the  first  figure  of  the 
quotient.  Multiplying  the  divisor 
by  this  quotient  figure,  and  sub- 
tracting the  product  from  470,  we 
have  116  for  a  remainder,  to  which 
we  annex  the  next  figure,  3,  of  the 
dividend,  thus  forming  the  number 
1163.  We  now  seek  how  many 
times  the  divisor  is  contained  in 
1163,  which  is  3  times.  We  place 
the  3  for  a  second  figure  of  the  quotient.  Multiplying  the  divisor  by 
this  second  figure,  and  subtracting  the  product  from  1163,  we  find 
101  for  a  second  remainder ;  to  which  annexing  5,  the  next  figure  of 
the  dividend,  we  have  1015.  Thus  we  proceed  till  all  the  figures  of 
the  dividend  have  been  brought  down. 

From  the  above  work  we  readily  deduce  the  following 
RULE. 

I.  Place  the  divisor  at  the  left  of  the  dividend,  keeping 
them  separate  by  a  curved  line. 

II.  Seek  how  many  times  the  divisor  is  contained  in  the 
fewest  figures  of  the  dividend  that  will  contain  it ;  set  the 
figure  expressing  the  number  of  times  at  the   right  of  the 
dividend  for  the  first  figure  of  the  quotient,  keeping  dividend 
and  quotient  separate  by  means  of  a  curved  line. 

III.  Multiply  the  divisor  by  this  quotient  figure,  and  sub- 
tract the  product  from  those  figures  of  the  dividend  used, 
and  to  the  remainder  annex  the  next  figure  of  the  dividend  ; 


§  28.]  DIVISION.  43 

then  find  how  many  times  the  divisor  is  contained  in  this  new 
number,  and  write  the  result  in  the  quotient. 

IV.    Continue  the  operation,  as  before,  until  all  the  figures 

of  the  dividend  have  been  brought  down. 

I 
NOTE  1. — Having  brought  down  a  new  figure,  if  the  number  thus 

formed  be  less  than  the  divisor,  it  will  contain  it  0  times ;  we  there- 
fore write  0  in  the  quotient,  and  bring  down  another  figure. 

NOTE  2. — If  in  multiplying  the  divisor  by  any  quotient  figure  we 
obtain  a  product  which  exceeds  the  number  we  sought  to  divide,  wo 
must  make  the  quotient  figure  smaller. 

NOTE  3. — If  a  remainder  should  be  found  larger  than  the  divisor, 
the  quotient  figure  must  be  taken  larger. 

PROOF. 

§  28.  The  dividend,  when  there  is  no  remainder,  is  a  com- 
posite number,  of  which  the  divisor  and  quotient  are  fac- 
tors. The  final  remainder,  if  any,  is  evidently  part  of  the 
dividend.  Consequently,  if  the  work  be  right,  the  product 
of  the  divisor  and  quotient,  with  the  remainder,  if  any,  add- 
ed, will  equal  the  dividend.  Or,  we  may  prove  or  test  the 
work  as  we  proceed,  by  adding  together  the  successive 
products  and  the  remainder,  if  any.  If  the  work  is  right, 
this  sum  will  equal  the  partial  or  entire  dividend,  as  the 
case  may  be.  Thus,  in  the  previous  example,  354,  the  first 
product,  and  1062,  the  second  product,  with  the  remainder, 
101,  when  added  together,  the  sum  is  equal  to  4703,  the 
partial  dividend,  &c. 

EXAMPLES. 

214-218.  Divide  826190  by  24;  48;  96;  112;  144; 
and  annex  the  remainders,  if  any,  to  the  quotients,  according 
to  §  26  ;  "  Note,"  under  the  rule. 

219-224.  Divide  9281746  by  27;  44;  98;  76;  236; 
294  ;  and  annex,  <fec. 


44  DIVISION.  [CHAP.  vi. 

225-235.  Dividend  829765149.  Divisors  486  ;  928  ; 
714;  386;  403;  907;  6172;  8316;  5793;  2405  ;  3006. 
Give  the  exact  quotients. 

236-243.  Divide  7200651897  by  2498  ;  76389;  32174; 
98263;  45208;  301987;  5678W  ;  890123. 

244-250.  Divide  8976014236  by  298701 ;  4853684 ; 
9130821;  1280319;  7600994;  3268753;  91465923. 

251-257.  Divide  123456789  by  789  ;  5763447  by 
678509  ;  1521808704  by  6503456  ;  243166625648  by 
3471032;  166168212890625  by  12890625  ;  11963109376 
by  109376  ;  24892456  by  36546. 

§  29.  When  the  divisor  ends  with  one  or  more  naughts. 
We  have  seen  (§  6)  that  a  number  is  multiplied  by  10,  by 
annexing  a  naught ;  by  100,  by  annexing  two  naughts,  &c. 
Conversely,  a  number  is  divided  by  10,  by  cutting  off  one 
naught  from  the  right ;  by  100,  by  cutting  off  two  naughts 
from  the  right,  &c. 

So  if,  instead  of  naughts,  significant  figures  are  cut  off 
from  the  right  of  a  number,  the  number  is  still  divided  by 
10,  100,  &c.,  while  the  figures  cut  off,  are  remainders  after 
the  division. 

Let  us  divide  2475  by  20. 

Having  cut  off  the  5  from  the  right  2|0  )  247 1 5 
of  the  dividend,  and  the  0  from  the 
right  of  the  divisor,  which  is,  in  ef- 
fect, dividing  both  dividend  and  di- 
visor by  10,  we  divide  247  by  2.  We  obtain  123  for  a  quotient  and  1 
for  a  remainder.  This  remainder  is  1  ten,  since  it  is  a  part  of  the  7 
of  the  dividend  which  occupies  the  tens'  place ;  annexing  the  5  units 
which  was  cut  off  to  the  1  ten  which  remained,  we  have  1  ten  and  5 
units,  or  15  for  the  true  remainder. 

NOTE. — This  case  may  be  comprised  under  that  wherein  the  divi- 
sor is  a  composite  number.  (See  forward,  §  29.)  Thus,  taking  the 


123    15  remainder. 


§  29.]  DIVISION.  45 

preceding  example,  the  divisor  20  =  2  X  10.  Dividing  2475  first  by 
10,  which  division  is  effected  by  cutting  off  the  right-hand  figure,  5, 
we  have  247  for  the  first  quotient,  and  5  for  the  first  remainder. 
Next,  dividing  247  by  2,  we  find  123  for  the  quotient  sought,  and  1 
for  the  second  remainder. 

Now,  by  the  rule  under  the  case  referred  to,  we  find  the  true  re- 
mainder to  be  1  X  10  +  5  =  15. 


Hence  the  following 


o 

RULE. 


(Jut  off  from  the  right  of  the  dividend  as  many  figures  as 
there  are  naughts  at  the  right  of  the  divisor ;  divide  what 
remains  by  the  divisor  without  the  naughts  at  its  right.  To 
the  final  remainder  annex  the  figures  cut  off  from  the  divi- 
dend, for  the  true  remainder. 

EXAMPLES. 

258-263.  Divide  by  20  tbe  following  numbers  :  1*7284  , 
365920;  9873542;  345678901;  135794680;  379653219. 

264-271.  Divide  69543218937  by;  240;  by  300;  by 
480  ;  by  700  ;  by  690  ;  by  4000  ;  by  80000  ;  by  900000. 

272-273.  Divide  7123545  by  421000;  1212121212  by 
42000. 

274-275.  Divide  123456789  by  12300;  7296148731 
2498000. 

276-279.  Divide  87369841  by  3000;  970000;  8103030; 
6090300. 

280-285.  Divide  943821900  by  tbe  following  numbers: 
78910  ;  36800  ;  42700  ;  9865200  ;  437001000  ;  9843020. 

286-289.  Divide  376549281  by  370x480;  by  630  X 
30^-10;  by  82X500-4-100;  by  8700x300-^-1000. 

When  a  divisor  is  a  composite  number,  it  may  sometimes 
be  convenient  to  divide  by  its  factors. 


46  DIVISION.  [CHAP.  vi. 


Thus,  if  we  wish  to  divide  944  by  105,  we 
may  resolve  the  105  into  its  factors,  3X5X7, 
and  divide  as  in  the  example.  The  only  dif- 
ficulty lies  in  obtaining  the  true  remainder  ; 
that  is,  the  remainder  which  would  have  re- 
sulted from  dividing  by  105  at  once. 


3  )  944 

5  )  314  2,  1st  rem. 

7  )  62  4,  2d  rem. 

8  6,  3d  rem. 


Since  each  unit  of  the  62  is  5  times  as  great  as  each  unit  of  314,  it 
follows,  that  each  unit  of  the  3d  remainder,  6,  which  is  a  part  of  62, 
is  also  5  tunes  as  great  as  each  unit  of  314.  Hence  the  remainder  6 
is  the  same  as  5  times  6,  or  30  units  of  the  same  kind  as  those  of  314 ; 
but  the  2d  remainder,  4,  being  a  part  of  314,  and  of  the  same  order, 
should  be  added  to  30,  making  34,  for  the  true  remainder  arising 
from  dividing  314  by  35  or  5X7.  Again,  since  each  unit  of  314  is  3 
times  as  great  as  each  unit  of  944,  it  follows,  that  each  unit  of  the 
34  is  also  3  times  as  great  as  each  unit  of  944.  Hence  the  remain- 
der 34  is  the  same  as  3  times  34  =  102  units  of  the  same  kind  as 
944 ;  but  the  first  remainder,  2,  being  a  part  of  944,  is  of  the  same 
order ;  so  that  102  +  2  =  104,  is  the  true  remainder  required. 

The  operation  may  be  arranged  thus :  (6X5+4)  X  3+2=1 04. 

Hence  the  following 

RULE. 

Divide  t?ie  given  sum  by  one  of  the  factors  of  the  divisor, 
and  the  quotient  thus  obtained  by  another  factor,  and  so  on. 
The  last  quotient  will  be  the  quotient  required. 

Multiply  the  last  remainder  by  the  divisor  preceding  the . 
last,  and  add  in  the  preceding  remainder  ;  multiply  this  sum 
by  the  next  preceding  divisor,  and  add  in  the  next  preceding 
remainder  ;  and  so  on. 

EXAMPLES. 

290-295.  Divide  8217  by  35  =  5x7;  33678  by  15  = 
5X3;  9591  by  72  =  9x4x2;  10859  by  49  =  7x7; 
926541  by  81  =  9x9;  987654  by  63  =  7x3x3.  Give 
the  true  remainders  of  the  preceding. 


§29.]  DIVISION.  47 


QUESTIONS  INVOLVING  THE  FOUR  GROUND-RULES. 

296.  A  person  owes  to  one  man  375  dollars,  to  another 
he  owes  708  dollars,  to  a  third  man  he  owes  911  dollars. 
How  much  does  he  owe  to  the  three  men  ? 

297.  A  farmer  has  sheep  in  five  fields:  in  the  first,  he 
has  917;  in  the  second,  249;    in  the  third,  413;  in  the 
fourth,  1QOO;  and  in  the  fifth,  he  has  197.     How  many 
sheep  has  he  in  the  five  fields  ? 

298.  A  person  owes  to  one  man  302  dollars,  to  another 
man  he  owes  707  dollars,  and  has  owing  to  him  2000  dol- 
lars.    How  much  will  remain  after  paying  his  debts  ? 

299.  A  farmer  receives  for  his  wheat  103  dollars,  for  his 
corn  60  dollars,  for  his  butter  511  dollars,  for  his  cheese 
1212  dollars,  for  his  pork  601  dollars.     He  pays  towards 
a  new  farm  1000  dollars,  for  a  new  wagon  50  dollars,  for 
hired  help  on  his  farm  290  dollars,  for  repairing  house  173 
dollars.     How  much  money  has  he  remaining  ? 

300.  A  person  wills  1200  dollars  to  his  wife,  300  dollars 
for  charitable  purposes,  and  what  remains  is  to  be  equally 
divided  among  6  children.     Allowing  his  property  to  amount 
to  8562  dollars,  how  much  would  each  child  have  ? 

301.  A  man  gave  13558  dollars  for  a  farm :  he  then  sold 
73  acres,  at  75  dollars  per  acre ;  the  remainder  stood  him 
in  at  59  dollars  per  acre.     How  many  acres  did  he  pur- 
chase ? 

302.  Four  boys  divide  336  apples  as  follows:  the  first 
takes-  one-sixth  of  the  whole ;  the  second  takes  one-fourth 
of  what  was  left ;  the  third  takes  one-half  of  what  was  then 
left ;  the  fourth  has  the  remainder.    What  number  of  apples 
did  each  boy  have  ? 

303.  Three  men  are  to  share  equally  in  the  sum  of  1236 
dollars.     How  many  dollars  will  each  have  ? 


48  DIVISION.  [CHAP.  vi. 

304.  Divide   1245  acres  of  land  equally  between  five 
brothers. 

305.  It  is  about  95000000  miles  from  here  to  the  sun. 
JSTow,  admitting  that  it  requires  8  minutes  for  light  to  "pass 
from  the  sun  to  the  earth,  how  many  miles  does  it  pass  in 
one  minute  ? 

306.  Allowing  22  bricks  to  be  sufficient  to  make  one  cu- 
bic foot  of  masonry,  how  many  cubic  feet  are  there  in  a 
work  which  requires  100000  bricks? 

307.  The  circumference  of  the  earth  is  about  25000  miles. 
How  long  would  it  require  for  a  person  to  travel  around  it, 
if  he  could  pass  uninterruptedly  at  the  rate  of  200  miles 
per  day? 

308.  In  1845  the  extent  of   post-roads  in  the  United 
States  was  143940  miles,  and  the  amount  paid  for  "the  trans- 
portation of  the  mail  during  the  same  year  was   2905504 
dollars.     How  much  was  the  average  expense  per  mile  ? 

309.  The   distance  of   Uranus   from   the   sun  is   about 
1860624000  miles.     How  many  hours  would  it  require  to 
pass  this  distance  at  1 8  miles  per  hour  ?     Also,  how  many 
days,  and  how  many  years,  counting  24  hours  to  the  day, 
and  365  days  to  the  year  ? 

310.  How  many  barrels  of  apples,  at  3  dollars  per  bar- 
rel, can  I  buy  for  2568  dollars?     And  if  one  tree  produce 
8  barrels,  how  many  trees  will  yield  the  required  amount  ? 

311.  An  estate  of  8100  dollars  was  divided  among  nine 
children  in  the  following  way :  the  first  had  100  dollars  and 
one-tenth  of  the  remainder ;  after  this  the  second  had  200 
dollars  and  one-tenth  of  the  residue ;  again,  the  third  had 
300  dollars  and  one-tenth  of  the  remainder,  and  so  on :  each 
succeeding  child  had  100  dollars  more  than  the  one  imme- 
diately preceding,  and  then  one-tenth  of  what  still  remained. 
What  was  the  share  of  each  ? 


§  29.]  DIVISION.  49 

312.  A.  and  B.  each  owe  C. :  A.  owes   1472  dollars, 
which  is  less  than  what  B.  owes  him,  and  yet  the  difference 
between  A.'s  and  B.'s  debts  is  719  dollars.     How  much 
does  B.  owe  C.  ? 

313.  Admitting  the  earth  to  move  68000  miles  per  hour, 
how  far  will  it  move  in  one  day  ;  and  how  far  in  a  year  of 
365  days? 

314.  If  the  President  of  the  United  States  expends  daily 
60  dollars,  how  much  will  he  be  able  to  save  at  the  end  of 
the  365,  out  of  his  salary  of  25000  dollars  ? 

315.  An  army,  consisting    of    4525  men,  have  1030*75 
loaves  of  bread.     At  the  end  of  21  days,  500  men  are  killed 
in  a  battle.     Now,  if  each  man  in  each  day  eat  one  loaf  of 
bread,  how  many  days  after  the  battle  will  the  bread  sus- 
tain the  army  ? 

316.  Two  locomotives  start  from  the  same  place,  and 
move  in  the  same  direction ;  the  first  goes  25  miles  each 
hour,  the  second  only  15  miles.     After  the  first  has  passed 
a  distance  of  100  miles,  it  commences  a  backward  motion, 
maintaining  the  same  velocity  until  it  meets  the  second  loco- 
motive.    How  many  hours  after  starting  will  they  meet  ? 
And  at  what  distance  will  they  meet  from  the  starting- 
point? 

317.  One  hundred  miles  of  railroad  track  are  to  be  laid 
with  heavy  rail,  requiring  116  tons  to  the  mile.     After  re- 
ceiving iron,  at  52  dollars  per  ton,  to  lay  58  miles,  the  price 
per  ton  was  increased  so  as  to  make  the  whole  cost  of  the 
entire  road  612944  dollars.     What  was  the  latter  price  per 
ton  of  the  iron  ? 

318.  A  person  bought  two  farms,  one  of  97  acres,  at  51 
dollars  per  acre,  and  the  other  of  111  acres,  at  47  dollars 
per  acre.     He  paid  9539  dollars  cash,  and  for  the  balance 
he  gave  5  horses.     What  were  the  horses  valued  at  ? 


50  GENERAL   PROBLEMS.  [CHAP.  VII. 

319.  A  person  having  a  salary  of  17 00  dollars  saves  970 
dollars  at  the  end  of  the  year.     How  much  on  an  average 
were  his  daily  expenses,  if  we  count  365  days  to  the  year  ? 

320.  A  man  travelled  832  miles  in  20  days :  during  the 
first  9  days  he  went  37  miles  daily ;  during  the  next  5  days, 
he  went  41  miles  daily.     How  many  miles  each  day  did  he 
travel  during  the  last  6  days  ? 


CHAPTER   VII. 
GENERAL  PROBLEMS  AND  PRINCIPLES 

PROBLEMS  FOUNDED  UPON  THE  FOREGOING  RULES. 

§  30.  a.  The  sum  of  two  numbers,  and  one  of  those  num- 
bers being  given,  to  find  the  other  number :  Subtract  the 
given  number  from  the  sum. 

NOTE. — Let  the  pupil  illustrate  by  an  example  this  and  each  sue1 
ceeding  problem. 

b.  The  difference  between  two  numbers  and  the  larger 
number  being  given,  to  find  the  smaller  number :  Subtract 
the  difference  from  the  larger  number. 

c.  The  difference  between  two  numbers  and  the  smaller 
number  being  given,  to  find  the  larger :  Add  the  smaller 
number  and  the  difference  together. 

d.  The  sum  and  the  difference  of  two  numbers  being 
given,  to  find  the  two  numbers  :   The  difference  added  to 
the  sum  will  give  twice  the  larger  number. 

NOTE. — Let  the  pupil  illustrate  and  explain ;  and  show  how  the 
Bmaller  number  may  be  found. 

e.  The  product  of  two  numbers,  and  one  of  those  num- 


§  30.]  GENERAL   PROBLEMS.  51 

bers  being  given,  to  find  the  other  number  :  Divide  the 
product  by  the  given  number. 

f.  The  dividend  and  quotient  being  given,  to  find  the  di- 
visor :  Divide  the  dividend  by  the  quotient. 

g.  The  quotient  and  divisor  being  given,  to  find  the  divi- 
dend :  Multiply  the  quotient  and  divisor  together. 


.  —  Let  the  pupil  be  required  to  illustrate  each  of  the  above 
problems. 

EXAMPLES. 

1.  The  sum  of  two  numbers  is  one  hundred  and  forty- 
seven  millions,  two  hundred  and  eight  thousand,  eight  hun- 
dred and  sixty-six  ;    one  of  the  numbers   is  twenty-three 
millions,  seven  hundred  and  eighty-five  thousand,  four  hun- 
dred and  thirty-two.     What  is  the  other  number  ? 

2.  A  gentleman  dying  bequeathed  387984  dollars  to  his 
two  children  :  one  obtained  44836  dollars.     What  was  the 
share  of  the  other  ? 

3.  The  difference  between  two  numbers  is  10144;  the 
minuend  is  69975.-    What  is  the  subtrahend? 

4.  The  difference  between  two  numbers  is  150110  ;  the 
subtrahend  is  729355.     What  is  the  minuend? 

5.  The  sum  of  two  numbers  is  1809004332  ;  their  differ- 
ence is  166304310.     What  are  the  two  numbers  ? 

6.  A  product  is  539902  ;  one  of  its  factors  is  23.     What 
is  the  other  factor  ? 

7.  Divide  the  sum  of  the  following  numbers  by  147  : 
31969217,  182681240,456703100. 

8.  A  person  was  desirous   of  knowing  the    amount  of 
money  bequeathed  to  each  of  two  children.     He  ascertained 
that  together  they  had  received  48465  dollars  ;    and  that 
one  had  received  20891  dollars  more  than  the  other.     How 
much  did  each  receive  ? 


52  GENEKAL   PROBLEMS.  [CHAP.  VII 

9.  The  product  of  two  numbers  is  374671924092  ;  one 
of  the  numbers  is  302014.     What  is  the  other  number  ? 

10.  A  dividend  is   4703598  ;    the    quotient   is   13287, 
What  is  the  divisor  ? 

11.  A  divisor  is  6503456;  the  quotient  is  234.     What 
is  the  dividend  ? 

12.  At  a  certain  election  the  whole  number  of  votes  re- 
ceived by  two  opposing  candidates  was  8737  ;  the  success- 
ful candidate's  majority  was  375.      How  many  votes  did 
each  receive  ? 

13.  If  the  distance  between  two  planets  is  239000000 
miles  when  they  are  on  opposite  sides  of   the  sun,  and 
49000000  miles  when  on  the  same  side  of  the  sun,  how 
far  is  each  from  the  sun,  if  we  suppose  their  orbits  perfectly 
circular  ? 

14.  In  an  orchard  of  1813  trees,  there  are  37  rows.    How 
many  trees  in  each  row  ? 

15.  A  certain  number  of  dollars  is  divided  among  365 
persons,  each  person  receiving  97  dollars.     How  many  dol- 
lars was  divided  ? 

16.  The  construction  of  13  miles  of  plank  road  cost  74334 
dollars.     How  much  was  that  per  mile  ? 

17.  In  a  field  of  maize  there  are  243  rows  of  187  hills  in 
each  row.     How  many  hills  in  all  ? 

18.  I  have  two  casks  of  wine,  which  together  contain  67 
gallons,  and  one  contains  17  gallons  more  than  the  other. 
How  many  gallons  does  each  contain  ? 

19.  Two  persons  starting  from  the  same  place  find  that 
when  they  travel  in  opposite  directions  they  are  at  the  end 
of  one  hour  18  miles  apart,  but  when  they  travel  in  the 
same  direction  they  are  at  the  end  of  one  hour  only  4  miles 
apart.     How  far  does  each  go  in  one  hour  ? 

20.  After  taking  371  dollars  from  a  certain  sum  of  money, 


§  31.]  GENERAL   PEOBLEMS.  53 

I  find  2*75    dollars   remaining.      How  many  dollars  were 
there  at  first  ? 

21.  Two  brothers  being  asked  their  ages,  the  elder  said 
he  was  59  years  old,  and  that  his  brother  was  just  15  years 
younger.     What  was  the  age  of  the  younger  ? 

22.  If  I  give  785  dollars  for  157  barrels  of  apples,  how 
much  do  I  pay  per  barrel  ? 

23.  How  many  barrels  of  apples  can  I  buy  for  785  dol- 
lars, at  5  dollars  per  barrel  ? 

24.  In  one  cubic  foot  there  are  1728  cubic  inches.     HOTT 
many  cubic  feet  are  there  in  174528  cubic  inches? 

25.  In  101   cubic  feet  there  are  174528  cubic  inches. 
How  many  cubic  inches  in  one  foot  ? 


PRINCIPLES  EVOLVED  FROM  DIVISION. 

§  31.  We  arc  now  prepared  to  understand  certain  im- 
portant relations  which  divisor,  dividend,  and  quotient  bear 
to  each  other. 

The  product  of  the  divisor  and  quotient  is  always  equal 
to  the  dividend.  Hence, 

a.  The  divisor  and  the  quotient  may  be  interchanged ; 
that  is,  if  the  dividend  be  divided  by  the  quotient,  the  result 
will  be  the  divisor ;  thus,  4)36(9  quot.,    9)36(4  quot. 

b.  If  the  divisor  remain  the  same,  multiplying  the  divi- 
dend by  a  given  number  has  the  effect  to  multiply  the  quo- 
tient by  the  same  number ;  thus,  36,  the  same  dividend  as 
above,  multiplied  by  2  and  divided  by  the  same  divisor,  4, 
will  give  18  for  a  quotient,  or  2  times  the  quotient  above. 

c.  If  the  dividend  remain  the  same,  multiplying  the  divi- 
sor by  any  number  has  the  effect  to  divide  the  quotient  by 
the  same  number.     Thus,  retaining  the  dividend,  36,  if  we 


54:  PRIME   AND  COMPOSITE  NUMBERS.       [CHAP.  VIH. 

multiply  its  divisor  by  3,  3x4  =  12,  the  quotient  is  3,  or 
the  same  as  9-^-3. 

d.  If  the  divisor  and  dividend  be  multiplied  by  the  same 
number,   there  will  be  no  change  in  the  quotient ;  thus, 
36x3  divided  by  4  x  3  will  give  9  for  the  quotient,  as  before. 

Still  further : 

e.  If .  the  divisor  remain  the  same,  dividing  the  dividend 
by  a  given  number  has  the  effect  to  divide  the  quotient  by 
the  same  number;  thus,  36-^3,  or  12  divided  by  4,  will 
give  9 -h  3,  or  3  for  a  quotient. 

f.  If  the  dividend  remain  the  same,  dividing  the  divisor 
by  any  number  has  the  effect  to  multiply  the  quotient  by 
the  same  number ;  thus,  4-i-2,  or  2  is  contained  hi  36,  9x2 
or  18  times. 

g.  If  dividend  and  divisor  be  divided  by  the  same  num- 
ber, the  quotient  will  remain  the  same ;  thus,  36-f-2,  or  18 
divided  by  4  -:-  2,  or  2  will  give  9  for  a  quotient  as  before. 


CHAPTER   VIII. 

PRIME  AND   COMPOSITE  NUMBERS. 
PRIME    NUMBERS. 

$  32.  A  COMPOSITE  NUMBER,  §  24,  is  one  which  can  be 
resolved  into  other  factors  besides  itself  and  units ;  thus,  12 
is  a  composite  number,  because  it  may  be  resolved  into 
2X3X2. 

A  prime  number  is  one  that  cannot  be  resolved  into  other 
factors  besides  itself  and  unity.  Thus,  1 1  is  a  prime  num- 
ber, because  no  two  or  more  numbers,  less  than  11,  and 
greater  than  1,  multiplied  together,  will  produce  11. 


§33.] 


PRIME  NUMBEKS. 


55 


Two  numbers  are  said  to  be  prime  to  eacli  other  when 
they  have  no  common  factor.  Thus,  8  and  9  are  prime  to 
each  other,  although  neither  is  a  prime  number. 

The  following  are  some  of  the  prime  numbers  : 


1 

29 

71 

113 

173 

229 

281 

349 

409 

463 

2 

31 

73 

127 

179 

233 

283 

353 

419 

467 

3 

37 

79 

131 

181 

239 

293 

359 

421 

479 

5 

41 

83 

137 

191 

241 

307 

367 

431 

487 

7 

43 

89 

139 

193 

251 

311 

373 

433 

491 

11 

47 

97 

149 

197 

257 

313 

379 

439 

499 

13 

53 

101 

151 

199 

263 

317 

383 

443 

503 

17 

59 

103 

157 

211 

269 

331 

389 

449 

509 

19 

61 

107 

163 

223 

271 

337 

397 

457 

521 

23 

67 

109 

167 

227 

277 

347 

401 

461 

523 

§  33.  The  pupil  may  be  aided  to  determine  whether  a 
number  is  prime  or  not  by  remembering  the  following  facts : 

1.  If  any  number  terminate  with  0  or  an  even*  digit,  the 
whole  will  be  divisible  by  2. 

2.  If  any  number  terminate  with  0  or  5,  the  whole  will 
be  divisible  by  5. 

3.  When  the  number  expressed  by  the  two  right-hand 
figures  is  divisible  by  4,  the  whole  will  be  divisible  by  4. 

4.  When  the  number  expressed  by  the  three  right-hand 
figures  is  divisible  by  8,  the  whole  will  be  divisible  by  8. 

5.  If  the  sum  of  the  digits  of  any  number  be  divisible  by 
3  or  9,  the  whole  number  will  be  divisible  by  3  or  9. 

NOTE. — "We  see  that  any  number  formed  by  a  succession  of  zeros 
placed  at  the  right  of  1,  as  10,  100,  1000,  <fec,,  will  contain  9  a  certain 
number  of  times  and  1  over.  If  the  numbers  are  formed  by  zeros 
placed  at  the  right  of  2,  as  20,  200,  2000,  <fec,,  9  will  be  contained  a 


*  An  even  digit  is  one  that  can  be  exactly  divided  by  2 ;  as,  2,  4,  6, 8.    An  odd 
digit  is  one  that  cannot  be  so  divided. 


56  PBIME  AND  COMPOSITE  NUMBERS.     [CHAP.  VIH. 

certain  number  of  times  and  2  over.  In  numbers  of  the  form  of  30, 
300,  3000,  <fcc.,  the  remainders  will  be  3  ;  and  so  on,  for  other  similar 
kind  of  numbers. 

Now  let  us  seek  the  remainder,  when  2846  is  divided  by  9.  The 
number  2846  is  made  up  of  2000,  800,  40  and  6.  2000  will  contain 
9  a  certain  number  of  times  and  2  over ;  800  a  certain  number  of 
times  and  8  over ;  40  a  certain  number  of  times  and  4  over.  If  then 
the  sum  of  these  remainders,  together  with  the  6  units,  will  exactly 
divide  by  9,  the  whole  number  is  divisible  by  9  ;  that  is,  a  number, 
is  divisible  by  9  when  the  sum  of  its  digits  is  divisible  by  9. 

The  same  will  be  true  of  3,  since  3  is  a  divisor  of  9  ;  that  is,  a  num- 
ber is  divisible  by  3  when  the  sum  of  its  digits  is  divisible  by  3. 

Find  the  prime  factors  of  868. 

We  see  at  a  glance  that  868  is  divisible  by 
2.  Again,  that  the  quotient  434  is  divisible 
by  2.  Dividing  the  second  quotient  217  by 
7,*  we  obtain  the  prime  quotient  31.  There- 
fore the  prime  factors  of  868  are  2,  2,  7,  31  ; 
so  that  2X2X7X31  =  868. 


EXAMPLES. 


2)868 

2)434 

7)"217 

31 


1-8.  What  are  the  prime  factors  of  12?  14?  15?  16? 
18?  20?  22?  24? 

9-16.  What  are  the  prime  factors  of  25?  26  ?  27  ?  28  ? 
30?  32?  33?  34? 

17-24.  What  are  the  prime  factors  of  35  ?  36  ?  38  ?  39  ? 
40?  42?  44?  45? 

25-32.  What  are  the  prime  factors  of  46  ?  48  ?  49  ?  50  ? 
51?  52?  54?  55? 

33-41.  Find  the  prime  factors  of  56  ;  57  ;  58  ;  60  ;  62  ; 
63  ;  64  ;  65  ;  66. 


*  Having  determined  by  inspection  that  a  number  is  not  divisible  by  2,  3,  5,  8, 
9,  or  10,  we  must  try  the  prime  numbers  in  succession,  beginning  with  7. 


§  34.]  GREATEST  COMMON  DIVISOK.  57 

42-50.  Find  the  prime  factors  of  68  ;  69;  70 ;  72  ;  85  ; 
87;  90;  96;  98. 

51-57.  Find  the  prime  factors  of  102  ;  111;  119;  125; 
138;  146;  155. 

58-63.  Find  the  prime  factors  of  154  ;  166  ;  178  ;  209  ; 
234;  259. 

64-69.  Find  the  prime  factors  of  309  ;  366;  375;  404; 
473;  524. 

70-76.  What  are  the  prime  factors  of  984?  of  1040? 
of  1368?  of  1224?  of  6584?  of  78903  ?  of  62148  ? 

77-83.  What  are  the  prime  factors  of  6918?  of  76540? 
of  63142  ?  of  78900  ?  of  6432  ?  of  97563  ?  of  89706  ? 

84-90.  Resolve  into  their  prime  factors  the  following 
numbers:  7498;  56234;  49750;  3333;  99939;  48765; 
92890. 

GREATEST  COMMON  DIVISOR. 

§  34.  The  numbers  12,  24,  48,  60,  can  each  be  divided 
by  2,  3,  or  6  ;  2,  3,  6  are  therefore  common  divisors  of 
those  numbers  ;  but  12  is  the  greatest  number  that  will  di- 
vide them  all;  therefore  12  is  their  greatest  common  di- 
visor. 

To  show  the  principle  of  this,  let  us  resolve  the  above  numbers  into 
their  prime  factors : 

12  =  2X2X3;  24  =  2X2X3X2;  48  =  2X2X3X2X2; 
60  =  2X2X3X5. 

Now,  evidently,  each  of  these  numbers  may  be  divided  by  one  of 
its  factors,  or  by  the  product  of  two  or  more  of  them.  Thus  each 
may  be  divided  by  2  ;  but  this  will  be  then*  least  common  divisor 
above  unity.*  So  each  may  be  divided  by  2  X  2. 

*  When  numbers  are  prime  to  each  other,  §  30,  they  have  no  common  divisor 
greater  than  unity. 


58  PRIME  AND  COMPOSITE  NUMBERS.    [CHAP.  VIET. 

f 

But  the  greatest  common  divisor  must  be  the  product  of  all  the  fac- 
tors common  to  all  the  numbers;  thus,  2X2X3=12,  the  greatest 
common  divisor. 

Take  another  example.  "What  is  the  greatest  common  divisor 
of  492,  744,  906?  492=2X2X3X41;  744=2X2X3X2X31; 
906=3  X  2  X 151.  The  factors  common  to  all  the  numbers  are  3  X  2  ; 
therefore  3X2=6  is  their  greatest  common  divisor. 


EXAMPLES. 

91-94.  What  is  the  greatest  common  divisor  of  360  and 
276?  of  592  and  599?  of  315  and  405  ?  of  1825  and  2655  ? 

95-99.  Find  the  greatest  common  divisor  of  506  and  308  ; 
of  404  and  364;  of  1112  and  616;  of  808  and  728;  of 
1518  and  924. 

100-103.  Find  the  greatest  common  divisor  of  492,  744, 
and  906  ;  of  246,  372,  and  522  ;  of  1476,  2232,  and  2718  ; 
of  738,  1116,  and  1566. 

104-106.  Find  the  greatest  common  divisor  of  252,  380, 
454,  and  500;  of  756,  1140,  1362,  and  1500;  of  1764, 
2660,  3178,  and  3500. 

107-109.  Find  the  greatest  common  divisor  of  632,  706, 
834,  and  8834  ;  of  1896,  2502,  2862,  and  1640 ;  of  3528, 
4424,  1942,  and  5164. 

Hence  we  have  the  following 

RULE. 

Resolve  each  number  into  its  prime  factors.  The  product 
of  all  the  factors  common  to  all  the  numbers  will  be  the 
greatest  common  divisor. 

§  35.  There  is  another  mode  of  determining  the  greatest 
common  divisor. 

a.  Any  number  that  will  exactly  divide  the  less  of  two 


§  35.]  GREATEST  COMMON  DIVISOR.  59 

numbers  and  their  difference,  will  also  divide  the  greater 
number.  Suppose  the  two  numbers  to  be  18  and  48.  6 
will  divide  the  less  number  18,  and  the  difference  30  ;  con- 
sequently it  will  divide  48.  For  the  number  of  times  the 
divisor  is  contained  in  the  less  number  plus  the  number  of 
times  it  is  contained  in  the  difference,  must  be  the  number 
of  times  it  is  contained  in  the  larger  number. 

b.  The  same  may  be  said  of  any  number  that  will  exactly 
divide  the  less  number  and  the  difference  between  any  num- 
ber of  times  the  less  number  and  the  greater  number.  Thus 
taking  the  number  above  :  18,  the  less  number,  x2  =  36  ; 
36  subtracted  from  48  leaves  12  ;  6  will  divide  the  remain- 
der 12,  and  the  less  number  18.  Consequently  it  will  di- 
vide the  larger  number  48. 

What  is  the  greatest  common  divisor  of  276  and  360  ? 

The  greatest  divisor  cannot  ex-          276  )  360  (  1 
ceed  the  less  number  276.    But  276 

276  will  not  divide  the  other  num-  

ber  360  without  a  remainder  84.  84: )  276  (  3 

Therefore  276  is  not  a  common  di-  252 

visor. 

Now  the  number  that  will  ex- 
actly divide  84  and  276  (a)  will 

also  divide  360.     We  seek  such  a  12  }  24  (  2 

divisor.  It  cannot  exceed  84. 
Trying  84,  we  find  it  will  not  di- 
vide 276  without  a  remainder,  24  ; 
84  is  therefore  not  the  greatest  common  divisor. 

Again,  the  number  that  will  exactly  divide  24  and  84  (6)  will  also 
divide  276.  This  divisor  .cannot  exceed  24.  But  24  will  not  divide 
84  without  a  remainder  12  ;  24  is  therefore  not  the  greatest  common 
divisor. 

Lastly,  the  number  that  will  exactly  divide  12  and  24  will  also 
divide  84  (6).  Trying  the  less  number  12,  we  find  it  exactly  divides 
the  other.  Therefore  12  is  the  greatest  common  divisor  of  276 
rend  360. 


60  PELME  AND  COMPOSITE  NUMBERS.       [~CHAP.  VIII. 

»  L 

Hence  the  following 

R'JLE. 

Divide  the  greater  number  by  the  less,  then  the  less  number 
by  the  remainder :  thus  continue  to  divide  the  last  divisor  by 
the  last  remainder,  until  there  is  no  remainder.  The  last 
divisor  will  be  the  greatest  common  divisor. 

NOTE. — If  the  greatest  common  divisor  of  more  than  two  numbers 
be  required,  find  first  the  greatest  common  divisor  of  t\9  >  of  them, 
then  of  the  divisor  so  found,  and  one  of  the  remaining  numbers,  and 
BO  on. 

EXAMPLES. 

110-117.  Find  the  greatest  common  divisor  of  365  and 
511  ;  of  115  and  161  ;  of  203  and  261  ;  of  145  and  185  ; 
of  120  and  350  ;  of  420  and  864  ;  of  560  and  768  ;  of  936 
and  1170. 

118-121.  Find  the  greatest  common  divisor  of  805  and 
1127  ;  of  1421  and  1827 ;  of  1015  and  1295  ;  of  888  and 
999. 

122-124.  Find  the  greatest  common  divisor  of  345,  483, 
and  609 ;  of  783,  435,  and  555  ;  of  2842,  3654,  and  2030. 

125-127.  Find  the  greatest  common  divisor  of  1602, 1603, 
1604;  of  311,  400,  510;  of  823,  800,  6?2. 

128-130.  Find  the  greatest  common  divisor  of  185,  259, 
407  ;  of  86,  430,  473  ;  of  505,  707,  4343. 

131-132.  What  is  the  greatest  common  divisor  of  2233, 
2030,  1827,  3045,  4060?  of  3885,  5550?  6105? 


LEAST  COMMON  MULTIPLE. 

§  36.  The  multiple  of  a  number  is  a  product  of  which  such 
number  is  a  factor.     Thus,  4,  6,  8,  are  multiples  of  2. 


§  37.]  LEAST  COMMON  MULTIPLE.  61 

A  common  multiple  of  two  or  more  numbers  is  a  product 
of  which  all  such  numbers  are  factors.  Thus,  48  is  a  com- 
mon multiple  of  2,  3,  4,  6,  8. 

The  least  common  multiple  of  two  or  more  numbers  is  the 
least  product  of  which  all  such  numbers  are  factors.  Thus, 
while,  as  above,  48  is  a  common  multiple  of  2,  3,  4,  6,  8, 
because  it  can  be  exactly  divided  by  each  of  those  numbers, 
24  is  their  least  common  multiple,  because  it  is  the  least 
number  that  can  be  so  divided. 

Evidently  a  number  or  a  series  of  numbers  can  have  an 
infinite  number  of  multiples. 

The  multiple  of  two  numbers  prime  to  each  other,  is  their 
product. 

Find  the  least  common  multiple  of  8,  16,  24. 

Let  us  resolve  these  numbers  into  their  prime  factors.  8=2  X  2  X  2 ; 
16=2X2X2X2;  24=2 X 2 X 2 X 3.  The  least  common  multiple  of 
these  numbers  must  contain  all  their  prime  factors  once.  We  then 
note,  first,  all  the  factors  of  8  :  2X2X2.  Next,  seeing  that  16  has 
an  additional  factor,  2,  and  that  24  has  an  additional  factor,  3,  we 
take  the  product  of  all  these  different  factors :  2  X2X2X 2X3=48, 
which  is  the  least  common  multiple  of  8,  16,  24.  "We  have  then  this 
rule,  for  finding  the  least  common  multiple  of  two  or  more  numbers. 

Resolve  each  number  into  its  prime  factors.  Multiply 
these  factors  together,  using  such  as  are  common  to  two  or 
more  numbers  BUT  ONCE.  The  product  so  found  will  be  the 
least  common  multiple. 

NOTE. — If  a  factor  be  used  more  than  once,  though  the  result  would 
be  a  multiple  of  the  given  numbers,  yet  it  would  not  be  the  least 
common  multiple. 

§  37.  It  may  be  sometimes  more  convenient  to  adopt  the 
following  method. 


62  PKIME  AND  COMPOSITE  NUMBERS.         [CHAP.  VIH. 

Find  the  least  common  multiple  of  10,  18,  21. 


We  divide  first  by  the  prime  factor  2, 


since  it  will  divide  two  of  the   numbers  3)5      9    21 

without  a  remainder,  and  place  the  quo- 


tients 5  and  9  together  with  the  undivided 
number  21,  in  the  line  below.     We  then 


2  )  10,  18,  21 


5,     3,     7 
2X3X5X3X7=630 


divide  by  another  prime  number  3,  for  the 
same  reason,  and  set  the  quotients,  with  the  undivided  number  5, 
below.  There  can  be  no  further  division  since  no  two  of  the  remain- 
ing numbers  have  a  common  divisor.  The  divisors  multiplied  into 
the  quotients  in  the  last  line,  will  give  the  least  common  multiple 
required. 

NOTE. — The  principle  of  this  operation  is  the  same  as  before.  Tit 
consists  in  finding  the  prime  factors  of  a  series  of  numbers  and  taking 
their  product.  Thus,  10=2X5;  18=2X3X3;  21=3X7.  2  being 
a  common  factor  of  10  and  18,  as  in.  the  above  divisor,  is  used  but 
once.  One  of  the  3's  being  common  to  9  and  21,  is  used  but  once 
as  in  the  second  division  above.  These  multiplied  into  the  5,  the  re- 
maining 3  and  the  7,  will  produce  the  result  sought.  Hence  the 
following 

RULE. 

Write  the  numbers  in  a  horizontal  line;  divide  them  by 
any  prime  number  which  will  divide  two  or  more  of  them 
without  a  remainder  ;  place  the  quotients  with  the  undivided 
numbers,  if  any,  for  a  second  horizontal  line  ;  proceed  with 
this  second  line  as  with  the  first ;  and  so  continue  until  there 
are  no  two  numbers  which  can  be  exactly  divided  by  the  same 
divisor.  The  continued  product  of  the  divisors,  and  of  the 
numbers  in  the  last  horizontal  line,  will  give  the  least  common 
multiple. 

133-138.  What  is  the  least  common  multiple  of  12,  16, 
and  24  ?  of  12,  15,  and  24  ?  of  11,  77,  and  88  ?  of  37  and 
41  ?  of  24,  60,  45,  180  ?  of  2,  4,  6,  8  ? 

139-143.  What  is  the  least  common  multiple  of  3,  5, 


§  38.]  CANCELLATION.  63 

7,  9  ?  of  2,  3,  4,  5,  6,  7,  8,  9  ?  of  7,  14,  16,  18,  24  ?  of  1, 
2,  3,  4,  5,  6,  7,  8,  9,  11  ?  of  12,  15,  16,  18,  20,  24? 

144-147.  What  is  the  least  common  multiple  of  36,  40, 
45,  60,  72,  90  ?  of  10,  20,  25,  50  ?  of  5,  9,  15,  18,  36,  135, 
1^2?  of  115,  184,  230,  460? 

148-151.  What  is  the  least  common  multiple  of  140,  168, 
210,  280,  420?  of  3,  5,  7,  14,  35,  42?  of  17,  19,  34,  38, 
209?  of  11,  13,  26,  99,  100? 

152-156.  What  is  the  least  common  multiple  of  8,  10,  12, 
13,  375?  of  34,  75,  88,  99?  of  2,  3,  5,  7,  11,  13,  17?  of 
4,  6,  8,  10,  19  ?  of  20,  21,  24,  48  ? 

CANCELATION. 

§  38.  Suppose  we  are  required  to  divide  35  tinus  99 
by  63.  As  these  numbers  are  composite,  we  have 

35X99     5X7X9X11       AT 

— = -. .     Now  we  know  (6  3 1 ,  g. )  that  divi- 

63  7x9  vy 

dend  and  divisor  may  be  divided  by  the  same  number  with- 
out altering  their  relation  to  each  other ;  in  other  words, 
without  affecting  the  value  of  the  quotient.  We  then  divide 
the  dividend  and  divisor  of  the  preceding  expression  by  7 

5X1  Xl  Xll 

and  by  9  ;  it  becomes — =  55.      We  may  per- 

1X1 

form  this  division  by  drawing  a  line  through  the  common 

5X^X0X11 

factors,  thus, — - — — ,  and  operating  upon  the   re- 

/*  X  $ 

maining  ones ;  the  expression  then  becomes =55. 

This  rejecting  of  common  factors  is  called  cancelation.  It 
is  a  great  saving  of  labor. 

When  all  the  factors  in  either  dividend  or  divisor  are  can- 
celed, write  1  in  their  place. 


64  PRIME  AND  COMPOSITE  NUMBERS.        [CHAP.  VIII 

Divide  the  product  21  times  22  times  65  by  1001. 
21X22X65_3X7X2XllX5Xl3_3X/X2X>'iX5Xly4_30_ 

Tool  7x~iixi3~  /rx]/xX~    ~~~^~ 

or  the  process  may  be  arranged  thus : 

First  cancel  the  factor  7  in  1001  and  in  21,  wri- 
ting the  other  factors  143,  and  3  below  and  above 
the  respective  numbers.  Next  cancel  the  11  in 
143  of  the  divisor  and  1 1  of  the  dividend,  writing 
the  other  factors,  13  and  2,  below  and  above  the  £$ 

respective  numbers  as  before.     Next,  cancel  the 
13  of  the  divisor  and  the  13  of  the  dividend,  writing  the  other  factor 

O  }S  Q  V>  K          O  A 

5  over  the  65.    The  expression  will  then  stand  -       — = — =30. 

EXAMPLES. 

157.  Divide  2X3X8X5X7  by  2x4x15. 

158.  A  man  had  34  filberts  in  each  of  49  different  piles. 
He  was  to  distribute  these  among  7  boys  and  7  girls.     How 
many  did  each  boy  and  girl  receive  ? 

159-160.  8X12  is  how  many  times  8  ?  -how  many  times 
12? 

161-162.  4  X  72  is  how  many  times  6  ?  how  many  times 
12? 

163-169.  72x48  is  how  many  times  6?  8?  12?  16? 
24?  32?  48? 

170-172.  What  is  the  value  of  36x100  divided  by 
10X18?  divided  by  2  X  20  ?  divided  by  9  X  10  ? 

173-175.  What  is  the  value  of  99x360x365  divided 
by  11  X  73  ?  divided  by  33  X  18  ?  divided  by  44  X  5  ? 

176-179.  What  is  the  value  of  33  X  77  divided  by  121  ? 
divided  by  21  ?  divided  by  7  ?  divided  by  3  ? 

180-183.  What  is  the  value  of  36x42x52  divided  by 
2X3X4?  divided  by  32X13?  divided  by  9  X  21  ?  divided 
by  21X13? 


§  39.]  FRACTIONS.  66 

184-187.  What  is  the  value  of  12X11X10  divided  by 
2x3x4?  divided  by  3  X  5  ?  divided  by  3x11?  divided 
by  4x5? 

188-190.  What  is  the  value  of  9x40x100  divided  by 
2X3X4X5?  divided  by  6X8X10?  divided  by  3  X  6  X  25  ? 


CHAPTER     IX. 

'    FRACTIONS. 

§  39.  A  FRACTION  is  a  part*  of  a  unit.  If  an  apple  be 
divided  into  2  equal  parts,  each  part  will  be  one-half  of  the 
apple  ;  that  is,  1  —  2,  or  J.  If  the  apple  be  divided  into 
3  equal  parts,  each  part  will  be-  one-  third  of  the  apple  ;  that 
is,  1-^-3,  or  ^,  &c. 

Suppose  3  apples  are  to  be  divided  among  5  boys.  Cut 
each  apple  into  5  equal  parts  or  fifths,  and  give  one  part  or 
fifth  of  each  apple  to  each  boy.  He  will  then  have  one- 
fifth  of  three  apples,  or,  what  is  the  same  thing,  three-fifths 
of  an  apple  :  in  figures,  |. 

We  see,  then,  that  the  number  of  parts  into  which  a  thing 
or  a  unit  is  divided,  is  expressed  by  the  figure  below  the 
line,  while  the  number  of  such  parts  as  are  taken  or  used  is 
expressed  by  the  figure  above  the  line. 

The  expression  f  may  be  read,  one-fifth  of  three  ;  or  3 
divided  by  5  ;  or  three-fifths.  The  latter  is  the  usual  mode. 

Read  the  following  expressions:  f,  ^,  r\,  T9T,  f£,  ££, 


The  number  above  the  line  is  called  the  numerator  :  the 

*  The  term  fraction  is  from  a  Latin  word  signifying  to  break,  meaning  a  broken 
part  of  a  unit. 

6* 


66  FRACTIONS.  [CHAP.  ix. 

number  below  the  line  is  called  the  denominator.  These  are 
also  called  the  terms  of  the  fraction. 

If  the  numerator  and  the  denominator  of  a  fraction  be 
equal,  the  value  of  the  fraction  is  unity  :  Jff=l.  If  an  ap- 
ple be  divided  into  12  equal  parts,  the  12  parts  or  twelfths 
will  make  the  whole  apple. 

If  the  numerator  be  less  than  the  denominator,  the  frac- 
tion is  called  a  proper  fraction ;  as  |,  ^,  J£. 

If  the  numerator  equal  or  exceed  the  denominator,  the 
fraction  is  called  an  improper  fraction  ;  as  -J,  if,  |f. 

When  a  whole  number*  and  a  fraction  are  connected,  the 
expression  is  called  a  mixed  number.  Thus,  4J,  3-i,  48T3T9T, 
are  mixed  numbers.  The  whole  number  is  called  the  inte- 
gral part  of  the  expression,  and  the  fraction  is  called  the 
fractional  part. 

A  fraction  of  a  fraction  is  called  a  compound  fraction. 
Thus,  i  of  f  of  f ,  f  of  I  of  |  of  ^i  f  of  |  of  f  of  f ,  &c., 
are  compound  fractions. 

Any  number  may  be  made  to  assume  the  form  of  an  im- 
proper fraction,  by  writing  under  it  a  unit  for  the  denomi- 
nator. Thus,  2,  3,  4,  5,  7,  &c.,  are  the  same  as  'j-,  %.,  -£,  ^, 

fcM 

Fractions  sometimes  occur,  in  which  the  numerator  or 
denominator,  or  both,  are  themselves  fractional ;  such  ex- 
pressions are  called  complex  fractions. 

Thus,  ~,  — ,  — f^,  — Y~,  &c.,  are  complex  fractions. 

A  fraction  is  said  to  be  inverted  when  the  numerator  and 
denominator  exchange  places.  Thus,  the  fractions  J-,  |-,  |-, 
To'  i>  f'  when  inverted,  become  f,  |,  f,  ^O-,  f,  f. 

These  fractions  are  called  COMMON  or  VULGAR  FRACTIONS, 

*  A  whole  number  is  also  called  an  Integer ;  thus,  5,  9,  24, 146,  are  integers. 


§  4:1.]  REDUCTION  OF  FRACTIONS.  67 

as  distinguished  from  another  kind,  to  be  hereafter  treated, 
called  Decimal  Fractions. 

$  40.  It  will  be  seen  that  Common  Fractions  are  founded 
upon  Division.  The  numerator  is  the  dividend,  the  denom- 
inator is  the  divisor.  The  fraction  itself  expresses  the,  quo- 
tient, or  the  value  of  the  quotient  resulting  from  the  division. 
Thus,  divide  6  by  3  ;  the  quotient  may  be  represented  by 
f =  2.  Divide  3  by  6  ;  the  quotient  -f,  three-sixths. 

From  the  relations  of  divisor,  dividend,  and  quotient,  as 
seen  in  §  31,  we  may  readily  infer  the  following 

PROPOSITIONS. 

I.  That,  multiplying  the  numerator  by  any  number  is  the 
fame  as  multiplying  the  fraction  by  the  same  number. 

II.  That,  multiplying  the  denominator  by  any  number  is 
the  same  as  dividing  the  fraction  by  the  same  number. 

III.  That,  multiplying  both  numerator  and  denominator 
by  any  number  does  not  alter  the  value  of  the  fraction. 

IV.  That,  dividing  the  numerator  by  any  number  is  the 
same  as  dividing  the  fraction  by  the  same  number. 

V.  That,  dividing  the  denominator  by  any  number  is  the 
same  as  multiplying  the  fraction  by  the  same  number. 

VI.  That,  dividing  both  numerator  and  denominator  by 
the  same  number  does  not  alter  the  value  of  the  fraction. 

REDUCTION  OF  FRACTIONS. 

§  41.  REDUCTION  is  the  process  of  changing  the  form  of 
an  expression  without  altering  its  value.  Thus,  the  integer 
1  may  be  reduced  to  the  fraction  f ;  the  fraction  j-f  may 
be  reduced  to  the  integer  1. 

Suppose  an  apple  be  cut  into  12  equal  parts :  6  of  those 


68  FRACTIONS.  [CHAP,  ix. 

parts  are  equal  to  one-half  the  apple  ;  that  is,  T%=i.  It 
will  be  seen  that  ^  is  the  result  of  the  division  of  the  nu- 
merator and  denominator  of  -£%,  by  6. 

There  is  evidently  the  same  relation  between  1  divided 
into  2  parts,  as  there  is  between  6  divided  into  12  parts. 
We  may  therefore  find  an  equivalent  value  of  a  fraction,  in 
lower  terms,  by  dividing  numerator  and  denominator  by  the 
same  number.  (§  40,  Prop.  VI.) 

Hence  to  reduce  a  fraction  to  its  lowest  terms,  Divide  its 
numerator  and  denominator  by  their  greatest  common  di- 
visor. 

Reduce  |f  f  to  its  lowest  terms.  The  greatest  common 
divisor  of  492  and  744  is  12.  §  34,  35.  Dividing  by  12,  we 
have  -|^-  for  the  answer. 

NOTE.  —  We  may  frequently  discover  numbers,  by  inspection, 
which  will  divide  both  numerator  and  denominator  without  a  re- 
mainder. When  this  is  the  case',  we  need  not  resort  to  the  rule  for 
obtaining  the  greatest  common  divisor,  until  we  have  divided  by 
such  numbers. 

EXAMPLES. 

1-5.  Reduce  to  then*  lowest  terms  |-  ;  J  ;  j3^  ;  ^T  ;  -J|-. 
6-11.  Reduce  to  their  lowest  terms  f  f  ;  f  f  ; 

ToT  '   "2To  • 

12-16.  Reduce  to  their  lowest  terms 


17-20.  Reduce   to   their   lowest    terms    g-||g- 
fff;  H^a 

21-25.  Reduce  to  their  lowest  terms  J£f-  ;  ||f  ;  fff  ; 

rHff 

26-30.  Reduce  to  their  lowest  terms  f  $|  ;  Jf  £  ; 


31-40.    Reduce    to    their   lowest   terms 

1764.  3378.   834  .  1896.  1640.3528.1942.   5826 

~23~G"o  '    TTTffcT  '   "SI'ST  '    2T  OT  '    2  ¥6"  2  '  TT2T  '  5T6T  '    T5T9"?1 


§  43.]  REDUCTION  OF  FRACTIONS.  69 

§  42.  To  reduce  an  improper  fraction  to  a  whole  or  mixed 
number. 

Reduce  -££-  to  a  whole  number. 

As  there  are  four-fourths,  -J,  in  a  whole  thing  or  unit,  in 
JL2-  there  will  be  as  many  whole  things  or  units  as  4,  the 
denominator,  is  contained  times  in  12,  the  numerator,  which 
is  3.  Therefore  ~V2-=3. 

Reduce  f  f  to  a  mixed  number. 

As  there  are  -}f  in  a  whole  one,  there  will  be  as  many 
whole  ones  in  f  f  as  13  is  contained  times  in  95,  which  is  7, 
and  4  thirteenths  over.  Therefore  jf  =7T\. 

It  will  thus  be  seen  that  the  division  expressed  by  an  im- 
proper fraction  may  be  actually  performed  as  in  division 
proper.  Hence  this 

RULE. 

Perform  the  division  expressed  by  the  fraction. 

EXAMPLES. 
41-50.  Reduce  to  whole  numbers  the  following  fractions  : 

--   --  2       a    s- 


55550 

51-64.  Reduce  to  mixed  numbers  ff  ;  -If-  ;  f  f  ; 

4A°-  ;  - 


-V-  ;  H1  -4°-  - 


65-68.  Reduce   to   a   whole   or  mixed  number 


76818 
Y5606"' 

§  43.  To  reduce  a  whole  or  a  mixed  number  to  an  im- 
proper fraction. 

Reduce  5  to  fourths.  In  a  unit  there  are  4  fourths.  In 
5  there  will  be  as  many  times  4  fourths  as  there  are  units ; 
that  is,  5  =  5X4  fourths  =• 


70  FRACTIONS.  [CHAP.  IX. 

Reduce  5f  to  fourths.     5  =-£-  as  before  ;  to  which,  if 
the  3  fourths  be  added,  the  sum  will  be  23  fourths  ;  that  is, 


The  denominator  of  the  required  fraction  being  given,  find 
fhe  numerator  by  the  following 


RULE. 

Multiply  the  whole  number  by  the  denominator  of  the 
fraction  to  which  it  is  to  be  reduced.  To  the  product  add 
the  numerator  of  the  fractional  part,  if  any. 

EXAMPLES. 

69-75.  Roduce  9  to  4ths ;  5ths  ;  6ths ;  7ths  ;  8ths  ; 
9ths  ;  lOths. 

76-86.  Reduce  to  improper  fractions  4^- ;  3  J ;  7f  ;  8^- ; 

87-96.    Reduce   to    improper  fractions    81ffJ ; 

:  "T'T! 


97-99.  Reduce  484T3¥6¥  to  an  improper  fraction  ; 
298||. 

§  44.  To  reduce  compound  fractions  to  simple  ones. 

Take  the  compound  fractions  f  of  -f-.  This  expression  is 
the  same  as  |-  taken  f  of  a  time,  or  f  X  J.  f  is  evidently 
2  X  J  :  the  expression  then  becomes  2  x  %  X  f  ;  £  of  f  is  the 

same  as  f-^3=— - -  (§  40,  Prop.  II.),  2  times =(Prop. 

•  X  o  7  X  o 

2x5      10 
I.)  - — ^=777-      Thus,   the  numerators   of   the  compound 

3x7      21 


§  44.]          REDUCTION  OF  FRACTIONS.  71 

fraction  have  been  multiplied  together  and  the  denominators 
together. 

Reduce  f  of  f  of  •£$  of  T7^-  to  a  simple  fraction. 

Canceling  the  factors  common  to  numerator  and  denomi- 
nator, the  expression  becomes 

3 

#     £      0      _7__      3x7      _  21 
3X5XI0X1?~5X5X4~TOO' 

5        4 

Hence  the  following 

RULE. 

First,  cancel  the  factors  common  to  the  numerators  and 
denominators  of  the  given  fraction ;  then  multiply  the  re- 
maining  numerators  together  for  a  new  numerator,  and  the 
remaining  denominators  together  for  a  new  denominator. 

NOTE. — When  a  fraction  is  to  be  multiplied  by  a  whole  number, 
the  whole  number  may  be  changed  to  an  improper  fraction  by  •wri- 
ting 1  for  its  denominator  ;  thus,  4=^. 

EXAMPLES. 

100-101.  Reduce  £  of  £  ;  T8g-  of  •£?  to  simple  fractions. 

102-109.  Reduce  to  their  simplest  forms  the  following 
fractions :  f  of  f|  ;  -|  of  £  of  T5T ;  J  of  f ;  f  of  J  of  f ; 
foffof  Jofi;f  ofif  ;  Mof  f  of  ft;  1  of  f  of  -tfL 

°fM- 

110-116.  Simplify  the  following  :  J  of  f  of  f  of  ^  ;  j 
off  off  of  if  of  |f;  ioff  ofl|;  l-off  of  41;  iof  § 
of  |  of  |;  f  off  of  |  off;  T9o  of  If  of  J  of  ff. 

117-121.  Simplify  f  of  &  of  f  of  jf  ;  ff  of  ^  of  if  ; 
4  Of  A  of  f|  of  21  ;  f  of  £f  of  |f  of  19  ;  f  of  if  of  f| 
of  62. 


72  FRACTIONS.  [CHAP.  ix. 

§  45.  To  reduce  fractions  to  a  common  denominator. 
We  know  (Prop.  III.)  that  the  value  of  a  fraction  is  not 
changed  by  multiplying  its  numerator  and  denominator  by 
the  same  number.  If,  then,  we  multiply  the  numerator  and 
denominator  of  each  of  a  series  of  fractions  by  the  product 
of  the  denominators  of  all  the  other  fractions,  we  shall  retain 
the  values  of  the  respective  fractions,  and  at  the  same  time 
they  will  have  a  common  denominator. 

Reduce  J,  J,  and  -f  to  a  common  denominator. 
The  numerator  of  i  becomes  1x3x7  =  21 
The  denominator  of  i    "         2x3x7  =  42 
The  numerator  off       "         5X2x7  =  70 
The  denominator  of  f    "         3x2x7  =  42 
The  numerator  of  f       "         4x2x3  =  24 
The  denominator  of-f-    "         7x3x2  =  42 
The  fractions,  then,  are  \ J,  \\,  \ f 
Hence  the  following 

RULE. 

Multiply  each  numerator  by  all  the  denominators  except  its 
own  for  a  new  numerator,  and  all  the  denominators  together 
for  a  common  denominator. 

NOTE. — Mixed  numbers  must  be  reduced  to  improper  fractions, 
compound  fractions  to  their  simplest  form,  and  all  the  fractions  to 
their  lowest  terms,  before  multiplying. 

EXAMPLES. 

122-129.  Reduce  to  common  denominators  %  and  f  ;  | 
and  f ;  f  and  | ;  f  and  &  ;  JJ  and  JJ. ;  ff  and  }£  ;  {$ 
and«;  «  and  if 

130-136.  Reduce  to  common  denominators  l,  J,  and  -*-; 
J,  i  and  i ;  j,  f ,  and  $  ;  f ,  J,  and  |  ;  TV  if  and  JJ ; 


§  46.]  DEDUCTION  OF  FKACTIO^S.  73 

137-139.  Reduce  to  common  denominators  J  of  f,  4J, 
51 ;  f  of  |,  of  5,  71  5i  ;  f  of  f ,  f  of  4j,  f  of  7f 

§  46.  When  the  Zeastf  common  denominator  is  required. 

Find  the  least  common  denominator  of  T5^,  T7g-,  -Jl.  The 
l^ast  common  multiple  of  12,  16,  24  (§  36),  is  48.  It  is 
evident  that  the  denominator  of  each  fraction  is  multiplied 
by  a  certain  factor  to  produce  this  multiple  :  that  is,  1 2  by 
1 ;  16  by  3  ;  24  by  2.  Now  if  the  numerator  of  such  frac- 
tion be  multiplied  by  the  same  factor,  each  fraction  will  re- 
tain its  value,  and  all  will  have  a  common  denominator ;  thus, 


2.0      2JL     22. 

48'    48'    48* 


Hence,  the  following 


RULE. 


Find  the  least  common  multiple  of  the  denominators  for 
the  least  common  denominator. 

For  each  new  numerator  multiply  the  numerator  of  each 
fraction  by  that  factor  of  the  multiple,  of  ivhich  the  denomi- 
nator of  such  fraction  is  the  other  factor. 

EXAMPLES. 

140-146.  Reduce  to  the  least  common  denominator  £,  -|, 
and  f  ;  J,  f ,  and  f  ;  f  and  f  ;  f  and  f  ;  f ,  f ,  and  f  ;  f , 
TVandif;f,|,andf. 

147-155.  Reduce  to  the  least  common  denominator  T5^, 

A, it;  i of  j  of  A,  A.  a*d  TV;  sMjhf ;  irVi*; 
A.  TV  e A  ;  i>  *•  H' and  i ;  A.  i  \,  A ;  f .  I'  f  f  A ; 

J'  i>  i'  I'  T9o>  TZQ- 

156.  Reduce  1,  l,  i,  i,  i,  1,  i,  l  to  equivalent  fractions 
having  a  common  denominator. 

7 


74:  FRACTIONS.  [CHAP.  IX. 

ADDITION  OF  FRACTIONS. 

§  47.  In  addition  of  whole  numbers,  we  have  seen  that 
units  can  only  be  added  to  units,  tens  to  tens,  &c.  3  units 
and  4  tens  will  make  neither  7  units  nor  7  tens  ;  but  4  tens 
=40  units;  and  40  units+3  units=43  units. 

So  in  fractions,  3  fourths  cannot  be  added  to  4  sixths,  for 
the  result  will  be  neither  7  fourths  nor  7  sixths.  But  3 
fourths  =  9  twelfths  and  4  sixths  =  8  twelfths  ;  and  9  twelfths 
and  8  twelfths  =17  twelfths,  or  1  and  5  twelfths;  that 
is>  f+|-=  T92+T82  =  H—  1lV  Hence,  for  the  addition  of 
fractions,  the  following 

,  RULE. 

JReduce  the  given  fractions  to  a  common  denominator.  Over 
this  denominator  place  the  sum  of  their  numerators. 

NOTE.  —  Seek  the  least  common  denominator  of  the  fractions.  If 
the  result  be  an  improper  fraction,  it  must  be  reduced  to  a  whole  or 
mixed  number. 

EXAMPLES. 

157-162.  What  is  the  sum  of  1,  i,  1,  and  l  ?  of  1  and 

i?  *,  TV  A?-rf*.  M?  of  j,  i  i  A?  off,  J,H? 

163-168.  What  is  the  sum  of  f  ,  },  f  ?  of  $,  |l,  if  ?  of 
«,  «,  «?  of  ||,  fi  A?  of  TV,  *,*?  off,  |,  /¥? 
169-174.  What  is  the  sum  of  1  J,  2j,  3}  ?  of  4J,  3j,  i  ? 


175-180.  What  is  the  sum  of  10J,  12^-,  15  J?  of  T3T, 
I?  of  fi  «,   |f?  of  ff,  |8,  i?    of    |,    4f    39    of 

^aV? 

181-186.  What  is  the  sum  of  2j,  4J,  6-f?  of  2^,  4J-, 
of  3J,  51   6}  ?  of  5J,  10TV,  15TV  ?  of  3J-,  61,  9-J  ?  of  2J 


§  48.]  SUBTRACTION  OF  FRACTIONS.  75 

SUBTRACTION  OF  FRACTIONS. 

§  48.  Subtract  \  from  -J.  This  cannot  be  done,  because 
the  fractions  have  different  denominators.  If  we  reduce 
both  to  thirty-fifths,  \  becomes  -f^,  and  J  becomes  ^  ;  and 
^j—-£j=^.  Hence,  to  subtract  one  fraction  from  an- 
other, the  following 

RULE. 

Heduce  the  fractions  to  a  common  denominator  :  over  this 
denominator  place  the  difference  of  the  numerators. 

EXAMPLES. 

187-]  95.  Find  the  difference  between  l  and  i  ;  f  and  f  ; 
t  and  T7T  ;  f  and  f  ;  TV  and  if  ;  |-  and  ii  ;  f  and  T8T  ;  | 
and  £f  ;  f  and  T8T. 

196-203.  Subtract  J  from  %  ;  \  from  \  ;  \  from  \  ;  J 
from  |  ;  i  from  il  ;  ^  from  TV  ;  H  from  }^  |  ;  T5^ 
from  ||^. 

204-209.  1  of  }  of  |-TV  -  5  f-4  of  J|=  ;  1  of 
3_iof|=  .  3i-2i^  ;  1  of  f  of  |-ioff^  ; 
f  of  |  off  of  f-f  of  |  of  |  off  =  . 

210-213. 


214-218.    What  is  the   value  of  (£+-J)  —  (}+J)  ?    of 
?  of        +-+?  of 


219-222.  What  is  the  value  of  (2j+3j)-(lJ+2j-)  ? 
?    of  of  I-        of         ?    of 


70  FRACTIONS.  [CHAP.  IX. 

MULTIPLICATION  OF  FRACTIONS. 

§  49.  Multiply  f  by  |. 

We  know,  §  44,  that  J  multiplied  by  £,  or  f  Xf,  is  the 
same  as  J  of  £.  Hence,  for  multiplication  of  fractions,  we 
must  use  the  same  rule  as  for  reducing  compound  fractions 
to  simple  ones. 

RULE. 

First,  cancel  the  factors  common  to  the  numerators  and 
denominators  of  the  given  fraction  ;  then  multiply  the  re- 
maining numerators  together  for  a  new  numerator,  and  the 
remaining  denominators  together  for  a  new  denominator. 

EXAMPLES. 

223-233.  Multiply  J  by  J  ;  J  by  J  ;  j-  by  f  ;  £  by  £  ; 
*l>y«;  f  byf;  i  by  tf  ;  £.  by  f  £  ;  J  by  f  f  ;  «  by 


234-240.  Multiply  together  i,  i,  and  i  ;  i,  J,  and  |  ; 
f  ,  |?  and  f  ;  f  ,  §-  ,  and  TV  ;  |,  -f-,  and  T6T  ;  f  ,  J,  and  ^f  ; 
f  ,  T\,  and  if. 

241-243.  What  is  the  product  of  f  of  |  by  ^  of  f  ?  of 
fofllbyfofH?  J  of  |  by  |  of  T^  of  1-1? 

244-246.  3jx4£xA=   ;  4^X3^=   ;  3ix4iX}=   . 

247-249.    Multiply  together  the  fractions  i,  |,   J,  f  ; 

f  i  i  -V-  ;  sj,  4j,  sj. 

250-254.  Multiply  together  ^  T7¥,  -f-  ;  f  by  4  ;  7  by  |  ; 
7Jby3j;  16jby5. 

255.  Multiply  the  sum  of  ^,  j,  J,  A,  by  the  sum  of  ^,  i, 


§  50.]  DIVISION  OF  INACTIONS.  77 

256.  Multiply  the  sum  of  -J  of  J,  f  of  f  by  the  sum  of 


257.  Multiply  f  of  |  of  f  of  T4r  by  £}  of  f  of  f  . 

258.  Multiply  the  sura  of  3,  3j,  3j,  3j,  by  the  sura  of 


DIVISION  OF  FRACTIONS. 

§  50.  Divide  ~  by  -|.  First  method.  If  the  fractions  be 
reduced  to  a  common  denominator,  their  numerators  may 
be  operated  upon  as  if  they  were  whole  numbers.  Thus, 
•£  =  !!•;  |-  =ff-.  And  §|  divided  by  |-|  is  the  same  as  32 
divided  by  35,  or  f  f  . 

Second  method,  -f-  divided  by  1  will  give  T  for  a  quo- 
tient; divided  by  -J-  (§3  1,/),  will  give  8  times  as  large  a 
quotient  as  when  divided  by  1,  or  yXy=-3y2-;  divided  by 
•f-  (§31,  c),  will  give  one-fifth  as  large  a  quotient  as  when 
divided  by  -J-,  or  ^  x—  ^-=  Jf  ,  the  same  result  as  found  by 
the  first  method,  we  see  that  in  fact  the  dividend  y  has  been 
multiplied  by  f  ;  that  is,  by  the  divisor  with  its  terms  in- 
verted. Hence,  for  dividing  one  fraction  by  another,  the 
following 

RULE. 

Invert  the  terms  of  the  divisor,  and  proceed  as  in  multipli- 
cation. 

NOTE.  —  If  either  dividend  or  divisor  be  a  whole  number,  make  it 
a,n  improper  fraction  by  giving  to  it  1  for  a  denominator. 

EXAMPLES 

259-269.  Divide  J  by  i  ;  \  by  £  ;  y  by  }  ;  j-  by  ^  ; 
A  by  A;  TWiVi  A  by  &;  fbyj;  fbyA;  |  by  f  ; 

I-  by  TV 

7* 


Y8  FRACTIONS.  [CHAP.  ix. 

27U-274.  Dividefbyi;  $  by  T'T  ; 


275-278.  Divide    4j  by  17±  ;    if  by  10  ;    \  of  £  by 
4-  of  |  ;  3i  of  2i  by  4}. 

279.  Divide  Jbyf  of  |. 

280.  Divide  the  sum  of  f  ,  £,  f  ,  f  ,  by  the  sum  of  1,  J,  J, 


281.  Divide  the  sum  of  f ,  f ,  *,  f ,  f ,  -^-,  ft,  if,  if,  if, 
by  the  sum  of  1,  J,  A,  i  j,  i,  i,  ^  ^  yL   j-L. 

282.  Divide  J  of  £  of  f  of  f  by  f  of  f  of  |  of  f . 
283-287.  Divide  the  sum  of  1,  Ij,  2J,  3j,  by  the  sum 

of  1J,  2^,  3^- ;  the  sum  of  ^  of  ^,  J  of  -|,  by  the  sum  of 
i-of  f,  iof|;  |  of  if  of  ff  byjof  f  of  f;  -I  of  if  of 
i  by  f  of  |  of  12  ;  i  of  f  of  f  of  £  by  J  of  J  of  8. 

RECIPROCALS. 

§  51.  The  reciprocal  of  any  number  is  found  by  dividing 
1  by  the  number.  Thus,  the  reciprocals  of  2,  3,  4,  are 

i-  i.  *• 

The  reciprocal  of  a  fraction,  for  example,  of  |-  is  l-±-^  = 
lxf=f.  Hence,  the  reciprocal  of  a  fraction  is  the  frac- 
tion inverted. 

Operations  in  division  may  therefore  be  included  under 
those  of  multiplication,  by  making  the  reciprocal  of  the  di- 
visor the  multiplier. 

EXAMPLES. 

288-295.  What  are  the  reciprocals  of  7,  8,  9,  11,  18,  24, 
96,  108? 

296-303.  What  are  the  reciprocals  of  f,  f,  |,  f,  f ,  f, 


§  51.]  FRACTIONS.  79 

304-309.  What  are  the  reciprocals  of  1^,  2j,  3j,  5|,  9f  , 
12T9r? 

310-313.  What  are  the  reciprocals  of  f  of  f-  ?  f  of  J? 
fof  ji?  f  off? 

314-318.  Perform  the  following  by  using  the  reciprocals 
of  the  divisors  :  f  -4-4  ;  T9o^f;  f-r-8;  4^7  ;  3j-r-2f 

MISCELLANEOUS  EXAMPLES  IN  COMMON  FRACTIONS. 

319-324.    Reduce    to  their  lowest  terms  £ff  ;    f  f  fj  ; 

.     15636.     505.     18999 
>    TF93  9-  >    5T5  '   ^TTTO" 

325-329.  Reduce  to  mixed  numbers  flf  ;    -?/-  ;    f  f  ; 


330-334.  Reduce  to  improper  fractions  3-|  ;  15-^-  ;  3T7T  ; 

;  IOOH- 

335-339.  Reduce  to  their  simplest  forms  \  of  f  of  f  ; 
f  of  |  of  |f  ;.£  ofi  of  A  of  3;  TV  of  f  of  fi  of  3j  ;  f  of 
jy*_  of  iof  100. 

340-344.  Reduce  i,  J,  j,  to  equivalent  fractions  having 
a  common  denominator  ;  so  \,  ^,  -J-,  -J,  |-  ;  3  J,  f,  f  ,  j3^-  ; 

111       1.35        7        11 
U>  T'  T'  TT  '    5>  T»  TT»  TTT- 

345-346.  What  is  the  sum  of  \,  i,  \  ?  of  f  ,  |,  f  ? 

347.  From  a  piece  of  cloth  -J  and  J-  of  the  whole  was  cut 
off.     What  part  of  the  whole  was  thus  taken  away  ? 
•     348-350.    From    \  subtract  \  ;    from  ^  subtract  ^  ; 
from  f  subtract  \. 

351.  A  tree  150  feet  high  had  -J  broken  off  in  a  storm. 
What  was  the  length  broken  off  ? 

352.  A.  and  B.  together  possess  1477  sheep,  of  which  A. 
owns  |-  and  B.  3..     How  many  belong  to  each  man  ? 

353.  A.  owns  T3T  of  a  ship,  valued  at  15422  dollars:  he 
sells  to  B.  f  of  his  share.     What  is  the  value  of  what  A. 
has  left  ;  also,  what  is  the  value  of  B.'s  part? 


80  FRACTIONS.  [CHAP.  ix. 

354.  A  cotton-mill  is  sold  for  30000  dollars,  of  which  A. 
owns  J  of  the  whole,  B.  and  C.  each  own  J  of  J  of  the  whole. 
How  many  dollars  does  each  one  claim  ? 

355.  A.  and  B.  have  a  melon,  of  which  A.  owns  f  and 
B.  f :  C.  offers  them  one  shilling,  to  partake  equally  with 
them  of  the  melon,  which  was  agreed  to.     How  must  the 
shilling  be  divided  between  A.  and  B.  ? 

356.  A  farmer  had  J  of  his  sheep  in  one  field,  £  in  a  sec- 
ond field,  and  the  residue,  which  was  779,  in  a  third  field. 
How  many  sheep  had  he  in  all  ? 

357.  If  I  divide  616  dollars  between  A.,  B.,  C.,  and  D., 
by  giving  A.  £  of  the  whole,  B.  T5T  of  the  remainder,  C.  | 
of  what  then  remained,  and  D.  the  balance,  how  much  will 
each  receive  ? 

358.  In  Fahrenheit's  thermometer  there  are  180  degrees 
between  the  boiling  and  freezing  points  ;  in  that  of  Reau- 
mur there  are  80.     What  fraction  of  a  degree  in  the  latter 
expresses  a  degree  of  the  former  ? 

359.  The  receipts  of  Jenny  Lind's  first  concert  in  New 
York  were  26000  dollars  ;  the  expenses  were  4000  dollars. 
Jenny  received  1000  dollars  as  her  regular  nightly  stipend, 
and  \  the  net*  proceeds  in  addition.     How  much  did  she 
receive  ? 

360.  Of  the  proceeds  of  her  first  concert  Jenny  Lind  do- 
nated as  follows  :  to  the  Fire  Department  Fund  T6^  ;  to  the 
Musical  Fund  Society  -^ ;  Home  for  the  Friendless  j1^- ; 
Society  for  Relief  of  Indigent  Females  y1^- ;  Dramatic  Fund 
Association  j1^  ;  Home  for  Colored  Aged  Persons  j1^  ;  Asy- 
lum for  Destitute  Females  j1^ ;  Orphan  Asylum  jL  ;  Ro- 
man Catholic  Orphan  Asylum  T^  ;  Protestant  do.  -f^ ;  Old 
Ladies'  Asylum,  the  remainder,  500  dollars.     How  much 

*  Net  or  neat  means  over  and  above  expenses. 


§  51.]  FRACTIONS.  81 

did  the  generous  singer  give  away  ?  and  how  much  did 
each  society  receive  ? 

361.  In  the  year  1850  there  were  probably  800000  bas- 
kets of  peaches  brought  into  New  York  city.  If  the  popu- 
lation were  510000,  what  fraction  will  express  how  many 
baskets  that  was  to  each  person  ? 

362-363.  A  journeyman's  wages  per  week  were  10  dol- 
lars :  of  that  sum  he  spent  \  for  the  6  working  days  for 
meat.  What  fraction  will  express  the  amount  he  spent  each 
day  ?  If  10  dollars  are  1000  cents,  how  many  cents  did  he 
spend  each  day  ? 

364.  Paid  137  dollars  for  flour,  at  6f  dollars  a  barrel. 
How  many  barrels  did  I  buy  ? 

365.  There  were  37  bushels  of  potatoes  in  a  cart :  J  of 
them  were  divided  among  3  families  of  4  persons  each  j  -J- 
of  them  among  2  families  of  7  persons  each  ;  j-  of  them 
between  3  persons;  and  the  remainder  among  18  persons. 
What  part  of  a  bushel  had  each  person  by  the  1st  division  ? 
What  part  had  each  by  the  2d  ?  what  by  the  3d  ?  what  by 
the  4th  ? 

366.  Two  persons,  A.  and  B.,  being  95  miles  apart,  travel 
towards  each  other,  both  starting  at  the  same  time.     They 
meet  at  the  end  of  6  hours,  when  they  discover  that  A. 
travelled  Ij  miles  more  than  B.  each  hour.     How  many 
miles  did  each  go  ? 

367.  A  boy,  after  losing  1  of  his  kite  string,  added  30 
feet,  and  then  found  that  it  was  just  |-  of  the  original  length. 
What  was  the  length  at  first  ? 

368.  A  person  commencing  business  with  a  certain  capi- 
tal, found  at  the  end  of  the  first  year  that  he  had  increased 
it  -J,  but  at  the  end  of  the  next  year,  having  been  unfortu- 
nate in  business,  his  capital  amounted  to  3000  dollars,  which 


82  DECIMAL  FRACTIONS.  [CHAP.  X. 

was  -J  of  what  he  had  at  the  end  of  the  first  year.     What 
was  the  capital  he  commenced  with  ? 

369.  A.  owns  \  of  J  of  \  of  a  ship  ;  B.  owns  \  of  £  of  f 
of  the  whole ;  C.  owns  J  of  f  of  \  of  the  whole  ;  D.  owns 
the  remainder.     How  much  does  A.'s  part  exceed  J  of  the 
whole  ?     How  much  does  B.'s  part  fall  short  of  \  of  the 
whole  ?     How  much  does  C.'s  part  fall  short  of  J  of  the 
whole  ?     How  much  does  D.'s  part  exceed  i  of  the  whole  ? 

370.  Divide   88  dollars  as  follows :  to  A.  give  1  dollar 
more  than  J  of  the  whole  ;  to  B.  give  10  dollars  more  than 
1  of  the  remainder ;  to  C.  give   14  dollars  more  than  \  of 
the  second  remainder  ;  and  to  D.  the  balance.     What  is 
each  one's  part  ? 


CHAPTER    X. 

DECIMAL   FRACTIONS. 

§  52.  SUPPOSE  1  to  be  divided  into  10  equal  parts  ;  each 
one  of  these  parts  is  1  tenth,  or  ^  ;  two  parts  are  2  tenths, 
or  T2Q-,  &c.  Now,  if  each  tenth  be  divided  into  ten  equal 
parts,  each  of  these  subdivisions  will  be  a  hundredth  ;  that 
*>  To--r-=  rVxA=T*ir-  So  ^+^.=JnfX&  = 

ToVo'  &c- 

Such  fractions  as  the  above,  which  decrease  or  increase 

only  in  a  tenfold  ratio,  are  called  Decimal*  Fractions.  So 
that  Decimal  Fractions  must  always  have  denominators  of 
the  following  form  :  10,  100,  1000,  10000,  &c. 

In  treating  of  whole  numbers,  we  saw  (§5)  that  the  suc- 
cessive orders  of  units  had  a  tenfold  increase  from  right  to 

*  Decimal,  from  a  Latin  word  signifying  ten. 


§  52.]  DECIMAL  FRACTIONS.  83 

left,  or  decrease  from  left  to  right.  This  being  true,  also, 
of  decimals,  they  may  be  written  down  and  operated  upon 
as  if  they  were  whole  numbers  ;  that  is,  their  denomina- 
tors may  be  omitted.  The  only  care  necessary  is  to  distin- 
guish the  decimal  from  the  integer  by  a  separatrix  or  point. 
Thus,  seven  and  3  tenths  is  written  7 '3  ;  six  and  9  hun- 
dredths  is  written  6 '09. 

The  first  place  at  the  right  of  the  decimal  point  is  tenths  ; 
the  second  place  is  hundredths  ;  the  third  place,  thousandths  ; 
the  fourth  place,  ten-  thousand ths  ;  the  fifth  place,  hundred- 
thousandths  ;  the  sixth  place,  millionths,  &c.,  as  in  the  fol- 
lowing 

TABLE. 


In  notation,  where  a  decimal  place  does  not  require  a  digit, 
0  must  be  written.  Thus,  3  hundredths  is  written  0*03,  the 
naught  showing  that  no  tenths  are  to  be  expressed  ;  7  thou- 
sandths is  written  O'OOY,  the  naughts  showing  that  no  tenths 
and  no  hundreds  are  to  be  expressed,  &c.  We  also  write 
a  naught  at  the  left  of  the  decimal  point  when  there  are  no 
units.  The  naught  is  thus  necessary  to  keep  the  decimal 
digit  in  its  proper  place. 

Every  naught  prefixed  to  a  decimal  carries  it  one  place 
further  to  the  right,  and  thus  decreases  its  value  10  times. 
Thus,  0-1=^;  0-01  =  ^;  0-001  =  ^,  &c. 

Naughts  annexed  to  a  decimal  do  not  alter  its  value,  since 


84:  DECIMAL    FRACTIONS.  [CHAP.  X. 

they  multiply  it.  5  numerator  and  denominator  by  the  same 
number.     Thus,  0-1=-^;  0-10  =  ^;  0'100=TWV 


§  53.  To  express  decimals  in  figures. 

Write  the  decimal  as  a  whole  number.  Prefix  as  many 
naughts  as  are  necessary  to  make  the  decimal  places  equal 
to  the  number  of  naughts  of  the  denominator.  Be  careful 
to  place  the  POINT  at  the  left  of  the  number. 

For  example  :  Express  in  figures  three  hundred  and  fifty- 
seven  millionths.  I  write  first  the  357.  There  are  6  naughts 
in  lOOOOOO^s  (T^IJWO)-  *  tnen  Prefix  to  tne  3  decimal 
places  already  written,  3  naughts,  0*000357. 

EXAMPLES. 

1-9.  How  many  decimal  places  in  1  hundredth  ?  in  1 
thousandth  ?  in  1  millionth  ?  in  1  ten-  thousandth  ?  in  1  hun- 
dred-thousandth ?  in  1  billionth  ?  in  1  ten-millionth  ?  in  1 
hundred-millionth  ?  in  1  ten-billionth  ? 

10-19.  Write  37  thousandths  ;  3  hundredths  ;  48  mil- 
lion/.hs  ;  95  hundred-millionths  ;  490  hundred-thousandths  ; 
1240  ten-millionths  ;  10000004  hundred-millionths  ;  96  bil- 
lionths  ;  9301  hundred-millionths  ;  27101  millionths. 

20-27.  Write  eight  hundred  and  four^  thousand  ten-mil- 
lionth^ ;  seven  million  and  four  hundred  millionths  ;  seven- 
ty-four million  and  eighty-one  billionths  ;  eight  hundred  and 
ninety-six  thousand  hundred-millionths  ;  four  thousand  and 
seven  hundred-thousandths  ;  eight  hundred  million  and  four 
thousand  ten-billionths  ;  sixty  billions  and  seventy-four  tril- 
lionths  ;  eight  hundred  billions  and  ninety-nine  ten-bil- 
lionths. 

NOTE.  —  The  teacher  will  exercise  the  pupils  in  similar  number^ 
until  they  can  write  them  with  rapidity  and  accuracy. 


§  54.]  DECIMAL  FRACTIONS.  85 

28-37.  Express  decimally  the  following  fractions  :  -f^Q  ; 
To  to" '  ToVo  o  '  ToVo'W  5  ToV/oV  5  nn$%Wo  >  ToWo  cflTO  '> 

12  .        1365 

o~¥o~o"o >  To oooo"- 


38-50.  Express  the  following  decimally  :  f£—  8T4Q-=8'4  : 


NOTE. — Perform  the  division  indicated  by  the  fraction. 

§  54.  To  read  decimals  expressed  in  figures. 
Read  the  figures  as  if  they  were  whole  numbers,  and  add 
the  name  of  the  right-hand  decimal  place. 

Thus,  0*7  is  read  seven  tenths ;  0*06  i  read  six  hun- 
dredths  ;  0*004  is  read  four  thousandths  ;  0*1070004  is  read 
one  million,  seventy  thousand  and  four  ten-millionths. 

NOTE. — If  the  pupil  numerate,  beginning  at  the  left,  thus,  "  tenths," 
"  hundredths,"  "  thousandths,"  <fcc.,  till  he  reach  the  last  figure,  he 
will  ascertain  the  name  of  the  right-hand  decimal  place. 

EXAMPLES. 

51-78.  Read  the  following  expressions  : 
0-8  0-10876  3-0017 

0-90  0-0001007  4-90018 

0-407         0-1000012  6-000001 

0-001         0-6750912         49-100007 
0-^945       0-80700176       86*0010007  36'21i 

0-87601     0-80000001       44-62000016  48*4081^ 

0-00076     0-901010101       O'lOOlOOOlOO 

An  expression  made  up  of  an  integer  and  a  decimal  is 
called  a  mixed  number  ;  as,  26-41. 

8 


86 


DECIMAL   FRACTIONS. 


[CHAP.  X 


ADDITION  OF  DECIMALS. 

55.  Add  7-8,  9*04,  78*005,  801*7604. 

We  arrange  the  numbers  so  that  tenths 
will  stand  under  tenths,  hundredths  under 
hundredths,  <fec.,  as  units  stand  under  units, 
tens  under  tens,  <fec.  We  then  add  as  in 
whole  numbers.  As  many  figures  must  evi- 
dently be  pointed  off  on  the  right  of  the  sum 
for  decimals,  as  are  equal  to  the  greatest 
number  of  decimal  places  in  any  of  the  num- 
bers added. 

896-6054 

Hence,  for  the  Addition  of  Decimals,  the  following 
RULE. 

I.  Place  the  numbers  to  be  added  so  that  the  figures  occu- 
pying the  same  decimal  place  shall  fall  in  the  same  column. 
Add  as  in  whole  numbers. 

II.  From  the  right  of  the  sum,  point  off  for  decimals  as 
many  figures  as  equal  the  greatest  number  of  decimal  places 
in  any  one  of  the  given  numbers. 

NOTE. — If  the  numbers  are  properly  set  down,  the  decimal  point 
in  the  sum  will  fall  directly  under  those  in  the  numbers  added. 

EXAMPLES. 

79-81.  Add  0-123,  0*012,  0*675,  and  0*0045;  0*14145, 
0*23235,  0*34345,  and  0*45455  ;  0*617,  9128,  76435,  and 
476280. 

82-87.  What  is  the  sum  of  0*3456,  0*3465,  0*3546, 
0*3564  ?  of  0*3645,  of  0*3654,  0*4356  ?  of  0*4365,  0*4536, 
0*4563?  of  0*4635,  0*4653,  0*5346,  0*5364?  of  0*6345, 
0*6354,  0*6435,  0*6453  ?  of  0*6534,  0*6543,  0*8765,  0'9876  ? 

88-93.  What  is  the  sum  of  1*234,  6*0045,  10*034?  of 


$  36.]  SUBTRACTION  OF  DECIMALS.  87 

0-0036,  0-01701,  0-4005  ?  of  37*38,  365'1,  63*36,  67*1  ?  of 
100-001,19-001,  48-5,  3*47?  of  12*2001,  21*012,  212-1, 
122-11  ?  of  401-104,  365,  390*91,  1000*1  ? 

94-98.  What  is  the  sum  of  256-7,  365-07,  I7'07l,  3*365? 
of  0-1924,  0-4501,  0'7512,  0*78301,  0'00019?  of  884-12, 
100-001,  303-044,  6'398,  48'485  ?  of  97l'914,  87*372, 
547-006,  533-014,  384-009  ?  of  203'145,  207*37,  0*017, 
0-099,  0-083  ? 

99-103.  What  is  the  sum  of  12078'5,  60075*8,  6'085, 
66-07,  301-38?  of  44'369,  27'036,  64'027,  125*125?  of 
105-317,  206-004,  6*001,  0'009,  0'478  ?  of  17*286,  3704, 
1076,  1710-1,  0-03457?  of  34689'14,  40057'82,  6078-65, 
47083-9,  34-567  ? 

-« 
§  56.     SUBTRACTION  OF  DECIMALS. 

RULE. 

Place  the  numbers  as  in  addition  of  decimals,  subtract  as 
in  whole  numbers.  Point  off  in  the  result  as  in  addition  of 
decimals. 

Subtract  0-000001  from  0*1. 

In  examples  of  this  kind  naughts  may  be  sup-      O'l 
posed  to  be  annexed  to  the  minuend,  which      O'OOOOOl 
(§  40,  Prop.  III.)  does  not  change  the  value  of      0*099999 
the  decimal. 

EXAMPLES. 

104-110.  From  898-7604  subtract  47-9631 ;  701-0001; 
37-2896;  0-4972;  1-0001;  897'6795  ;  2-461. 

111-116.  From  92581*31  subtract  8461*1;  94-0009; 
0*655816;  82000;  0*000036;  41*498. 

117.  From  3  millions  and  1  millionth  subtract  1  tenth. 


88  DECIMAL  FRACTIONS.  [CHAP.  X. 

118.  From  96  billions,  2  thousand  and  7,  subtract  84  ten 
millionths. 

119.  From  82  millions  3  hundred,  subtract  7  and  9  hun- 
dred-thousandths. 

120-122.  From  345-345  subtract  54-123;  from  1245*3478 
subtract  340*0122;  from  3456-12347846  subtract  479' 
100345. 

123-125.  Subtract  99'9  from  1023'4  ;  0'13047  from 
0-4785  ;  0-00675  from  0-11232. 

126-129.  Subtract  10'9807  from  219*307  ;  365*365 
from  4017-37  ;  301-627  from  505'0005  ;  404*3737  from 
900-1301. 

MULTIPLICATION  OF  DECIMALS. 

§  57.  A  tenth  taken  once  must  give  1  tenth  for  a  product  ; 
if  taken  only  one-tenth  of  a  time,  the  product  will  be  one- 
tenth  of  a  tenth,  or  one-hundredth  ;  that  is,  ro  xTo—  rio^* 
or  decimally  expressed,  O'l  X0'l=0-01.  This  is  evidently 
true,  since  if  the  tenth  part  of  any  thing  be  divided  into  10 
equal  parts,  each  subdivision  will  be  a  hundredth  part  of 
the  whole.  So  ^  of  r£o  =TWo-  and  so  on- 

Multiply  0*136  by  0*78.  If  we  supply  the  denominators 
of  these  decimals,  which  denominators  are  always  under- 
stood, we  shall  have  O'l  36  =TVo%  J  O^S^ 

Hence,  multiplying  ^fifo  by  ^,  we  find 

X         = 


From  which  we  see  that  the  number  of  decimal  places  in 
the  product,  always  denoted  by  the  number  of  naughts  in 
the  denominator  which  is  understood,  is  equal  to  the  num- 
ber of  decimal  places  in  both  factors.  Hence  we  have  this 


§  58.]    '  MULTIPLICATION  OF  DECIMALS.  89 

RULE. 

Multiply  as  in  whole  numbers.  From  the  right  of  the 
product  point  off  as  many  figures  for  decimals  as  there  are 
decimal  places  in  both  the  factors.  If  there  be  not  enough 
figures  in  the  product,  prefix  naughts. 

Multiply  0-125  by  0'37. 

In  this  example,  the  multiplicand  has  3  deci- 
mal places,  and  the  multiplier  2.  Therefore  the 
product  must  have  five  places.  And  since  there 
are  but  4  figures  in  the  product,  we  prefix  1 
naught  before  placing  the  decimal  point. 

EXAMPLES. 

130-138.  Multiply  943-078  by  8;  by  12;  by  14  ;  by  28; 
by  39  ;  by  121  ;  by  696  ;  by  1240  ;  by  67932. 

139-147.  Multiply  270-4601  by  2'1;  7'09;  6'003 ; 
92-804;  0-073*0-2946;  0-94820;  0-765921 ;  1023'6921. 

148-154.  Multiply  0-49801  by  36-296;  492*12  ;  37'009; 
6-219786  ;  0*000006  ;  0*0000009  ;  0-000000008. 

155-163.  Multiply  0'00074  by  0*19;  0'028;  0*0036; 
0*00048  ;  0-000096  ;  0*0000084  ;  0*00000097  ;  3648*1  ; 
8936*0004. 

§  58.  A  decimal  number  may  be  multiplied  by  10,  100, 
1000,  &c.,  by  removing  the  decimal  point  as  many  places 
to  the  right  as  there  are  naughts  in  the  multiplier.  If  the 
number  do  not  contain  so  many  figures,  annex  naughts. 

EXAMPLES. 

164.  Multiply  82*146  by  10. 

165.  Multiply  76*92  by  1000. 

166-171.  Multiply*  4610-4  by  10;  100;  1000;  10000; 
100000:  1000000. 

8* 


90  DECIMAL  FRACTIONS.  [CHAP.  X. 

172-179.  Multiply  0-47692  by  10;  100;  1000;  10000; 
100000;  1000000  ;.  10000000  ;  100000000. 

180-187.  Multiply  3-7  by  10 ;  100;  1000;  10000; 
100000;  1000000;  10000000;  100000000. 


DIVISION  OF  DECIMALS. 

§  59.  In  multiplication  of  decimals,  we  know  that  the 
number  of  decimal  places  in  the  product  is  equal  to  the  sum 
of  those  in  both  the  factors.  Now,  since  the  product  di- 
vided by  one  of  the  factors  must  produce  the  other  factor 
or  quotient,  it  follows  that  in  division  the  decimal  places  of 
the  dividend  must  be  equal  to  the  number  of  places  in  both 
divisor  and  quotient.  Hence,  the  number  of  decimal  places 
in  the  quotient  must  equal  the  excess  of  those  in  the  divi- 
dend over  those  in  the  divisor. 

Divide  5*81224  by  5'432. 

Dividing  581224  by  5432,  we  find  107  for  the  quotient. 
Since  5  figures  of  the  dividend,  and  only  3  figures  of  the 
divisor  are  decimals,  it  follows  that  two  figures  of  the  quo- 
tient 107  must  be  decimals,  so  that  1'07  is  the  quotient 
sought.  Hence  the  following 

RULE. 

Divide  as  in  whole  numbers  ;  point  off  as  many  decimal 
places  in  the  quotient  as  those  in  the  dividend  exceed  those 
in  the  divisor  ;  if  there  are  not  as  many,  supply  the  defi- 
ciency by  prefixing  naughts. 

NOTE. — Division  of  Decimals  may  be  explained' as  follows  : 

Suppose  dividend  and  divisor  to  be  whole  numbers,  the  quotient 

will  be  a  whole  number.     If  the  dividend  be  divided  by  10,  that  is, 

if  it  contain  one  decimal,  the  quotient  (§  3l,  e)  will  be  divided  by 

10  ;  that  is,  it  will  contain  one  decimal ;  and  generally  as  many  times 


§  60.]  DIVISION  OF  DECIMALS.  91 

as  the  dividend  is  divided  by  10  will  the  quotient  be  so  divided. 
But  if  the  divisor  also  be  divided  by  10,  the  quotient  just  obtained 
will  be  multiplied  by  10  (§  31,  /),  and  in  general  as  many  times  as 
the  divisor  is  divided  by  10,  so  many  times  will  the  quotient  be  mul- 
tiplied by  10 ;  that  is,  for  every  decimal  place  in  the  divisor  one  de- 
cimal place  in  the  quotient  must  be  cancelled. 

It  is  thus  seen  that  in  the  effect  upon  the  quotient,  each  decimal 
place  in  the  divisor  cancels  a  decimal  place  in  the  dividend,  and  that 
the  excess  of  decimal  places  in  the  dividend  over  those  in  the  divi- 
sor, that  is,  the  number  of  uncancelled  10's  by  which  it  is  divided, 
must  be  expressed  by  the  same  number  of  decimal  places  in  the 
quotient. 

Divide  0-123428  by  11'8. 

In  this  example,  tbe  divi-         ii-$  )  0*123428  (  0*01046 


dend  contains  6  decimal 
places,  and  tbe  divisor  but  1  ; 
the  quotient  must,  therefore, 
contain  5.  As  there  are  but 
4  figures  in  the  quotient,  sup- 


118 

542 

472 


708 
708 


ply  the  deficiency  by  prefix- 
ing a  naught  before  placing  the  decimal  point. 

EXAMPLES. 

188-192.  Divide  7'11  by  3'1  ;  24-06  by  8'02  ;  67*2336 
by  6*003  ;  96*97662  by  37*2987  ;  2146-078488  by  37*84. 

193-198.  Divide  3'810688  by  1'12  ;  0'109896  by  0*241  ; 
1-12264556  by  1-0012  ;  Q'01764144  by  0*0018  ;  0*07056545 
by  0-0073  ;  0-1411309  by  0'00365. 

§  60.  When  there  are  not  as  many  decimal  places  in  the 
dividend  as  in  the  divisor,  naughts  may  be  annexed  (§  40, 
Prop.  III.)  to  the  dividend.  When  the  number  of  decimal 
places  is  the  same  in  dividend  and  divisor,  the  quotient  will 
be  a  whole  number.  Thus,  j%-^T2^~f =3  '•>  ^iafc  'ls>  °'6^- 
0-2  =  3. 


92  DECIMAL  FRACTIONS.  [CHAP.  X. 

EXAMPLES. 

199-205.  Divide  0'7  by  0'07  ;  0'25  by  O'OOOo  ;  0'25  by 
0-00005;  0'125byO-000005;  122-418  by  3*4005  ;  244-431 
by  1-2345;  365' 2  by  9'13. 

206-213.  Divide  234-31  by  0'4967  ;  by  0'28160  ;  by 
2-00076;  by  7'892165  ;  by  22*872003;  by  41'9865432; 
by  221-762Q80  ;  by  3-4076321. 

214-225.  Divide  827640-32167  by  8'2  ;  by  9'03  ;  by 
11-416;  by  327-0489;  by  7260-19876  ;  by  9831-00014; 
by  63-222219;  by  92-4234767;  by  38'9l765890;  by 
21814-26;  by  8'4  ;  by  9*701. 

NOTE. — The  annexing  of  naughts  to  the  dividend  is  obviously  to 
reduce  dividend  and  divisor  to  a  common  denominator. 

Where  the  decimal  places  of  the  divisor  are  fewer  than 
those  of  the  dividend,  naughts  are  always  supposed  to  be 
annexed  to  the  divisor;  thus,0-8215-f-0'5  =  0-8215~0-5000. 
Of  course,  then,  if  the  dividend  contain  the  divisor,  the  first 
figure  of  the  quotient  will  be  a  whole  number. 

§  61.  When  there  is  still  a  remainder,  we  may  continue  to 
annex  naughts  to  it  and  to  divide,  until  a  sufficiently  accurate 
result  is  obtained.  The  sign  -f-  annexed  to  the  quotient  shows 
that  it  is  larger  than  is  written. 

NOTE. — The  pupil  will  remember  that  every  naught  annexed  to  a 
remainder  adds  another  decimal  place  to  the  dividend. 

EXAMPLES. 

226.  Divide  0'8215  by  0'5. 

227-231.  Divide  4-1175  by  0-5  ;  by  25  ;  by  35  ;  by  45  ; 
by  55. 

232-238.  Divide  20  by  0'003  ;  37'4  by  4'5  ;  7'85  by 
3-43  ;  0-478  by  0'58  ;  0'9009  by  0'405l  ;  68'283  by  9'22  ; 
845-6501  ;  by  37'37. 


§  62.]  DECIMAL  FRACTIONS.  93 

§  62.  To  divide  a  decimal  by  10,  100,  1000,  &c. 

T*o--T°—T*o  X  rV^ToVo  ;  that  is>  O'Ol -0-1  =  0-001. 
Hence  the  following 

RULE. 

Remove  the  decimal  point  as  many  places  to  the  left  as 
there  are  naughts  in  the  divisor  :  when  there  are  not  figures 
enough  in  the  dividend  prefix  naughts. 

EXAMPLES. 

239-242.  Divide  41497'6  by  10;  by  100;  by  1000; 
by  10000. 

243-247.  Divide  67'4  by  10;  by  100;  1000;  10000; 
100000. 

248-253.  Divide  0-341  by  10;  100;  1000;  10000; 
100000;  1000000. 

PROMISCUOUS    EXAMPLES    IN    DECIMALS. 

254.  Bought  4  loads  of  wood :  the  first  contained  0'97 
cords,  the  second  contained  1*03  cords,  the  third  contained 
0'945  cords,  the  fourth  contained  T005  cords.     What  did 
the  four  loads  measure  in  decimals  ? 

255.  In  the  month  of  May  the  amount  of  rain  was  3*15 
inches,  in  June    it  was  4'05  inches,  in    July  it  was  2 '9*7 
inches,  and  in  August  it  was  3 '03  inches.     How  much  rain 
fell  during  these  four  months  ? 

256.  During  three  successive  days  the  mean*  range  of 


*  If  the  sum  of  a  scries  of  unequal  quantities  be  divided  by  the  number  of  quan- 
tities, the  quotient  is  called  the  mean  or  average  of  these  quantities,  since  it  will, 
when  repeated  as  many  times  as  there  are  unequal  quantities,  just  equal  their  sum. 
Thus,  the  average  of  2,  4,  6,  8,  and  10  (5  quantities),  is  (2  -f  4  +  G  +  8  +  10)  -*-  5 

:_;  30  4-5=0. 


94  DECIMAL  FRACTIONS.  [CHAP.  X. 

the  barometer  was  29*04  inches,  29*51  inches,  and  29*73 
inches  respectively.     What  is  the  sum  of  these  heights  ? 

257.  In  1844,  the  whole  number  of  school  districts  of 
New  York  was  10990,  and  the  whole  number  of  children 
in  said  districts,  between  the  ages  of  5  and  16  years,  was 
696548.     What  was  the  average*  number  for  each  district  ? 

258.  In  New  York,  the  total  number  of  volumes  in  the 
11018  school-district  libraries  was  1145250.     What  was  the 
average  number  for  each  library  ? 

259.  In  one  mile  there  are  1760  yards,  and  in  one  rod 
there  are  5-J  =  5*5  yards.     How  many  rods  in  one  mile  ? 

260.  If  light  passes  191515  miles  in  a  second,  how  many 
seconds  will  if  require  to  pass  from  the  sun  to  the  earth,  a 
distance  of  95500000  miles  ? 

261.  If  a  cubic  inch  of  pure  water  weigh  252*458  grains 
avoirdupois,  of  which  7000  make  one  pound,  what  is  the 
weight  of  the  Imperial  or  English  gallon,  which  contains 
277*274  cubic  inches  ? 

262.  If  one  Imperial  gallon  contain  277*274  cubic  inches, 
how  many  cubic  inches  in  8  gallons  or  one  bushel,  and  how 
many  cubic  feet  of  1728  inches  each  ? 

263.  If  one  cubic  inch  of  pure  water  weigh  252*458  grains 
avoirdupois,  how  many  grains  will  1728  cubic  inches,  or 
one  cubic  foot,  weigh,  and  how  many  pounds  of  7000  grains 
each  ? 

264.  If  at  each  stroke  of  the  piston-rod  of  a  locomotive 
engine  a  distance  of  13*25  feet  is  passed  over,  how  many 
strokes  must  be  made  in  passing  a  distance  of  93  miles  ? 

265.  In  one  mile  there  are  5280  feet,  and  in  one  rod 
there  are  16*5  feet.     How  many  rods  in  one  mile  ? 

266.  How  many  feet  in  circumference  must  a  wheel  be 

*  See  note  on  preceding  page. 


§  63.]  FRACTIONS  TO  DECIMALS.  95 

so  as  to  roll  over  just  100  times  in  going  a  distance  of  one 
mile? 

267-269.  If  the  circumference  of  the  forward  wheel  of 
a  carriage  is  15 '25  feet,  and  the  circumference  of  the  hind 
wheel  17'75  feet,  then  in  a  journey  of  10  miles,  how  many 
times  will  each  revolve  ?  and  how  many  more  times  will 
the  one  revolve  than  the  other  ? 

270.  If  37-03  acres  of  land  cost  2000  dollars,  how  much 
was  it  per  acre  ? 

271-274.  If  I  purchase  43'25  acres  of  land  at  55'5  dol- 
lars per  acre,  and  sell  31 '25  acres  for  2500  dollars,  then  how 
much  did  I  give  for  the  whole  ?  How  much  did  I  receive 
per  acre  for  what  I  sold  ?  How  much  more  did  I  receive 
for  what  I  sold  than  the  whole  cost  me  ?  and  how  many 
acres  remained  unsold  ? 

275.  From  a  cistern  containing  3000  gallons,  73'5  bar- 
rels, of  31'5  gallons  each,  are  drawn  off.  How  many  gal- 
lons remain  ? 


REDUCTION  OF  COMMON  FRACTIONS  TO  DECIMALS. 

§  63.  Reduce  f  to  a  decimal. 

We  cannot  divide  3  by  8  ;  but  reducing  the  3  to  tenths, 
that  is,  multiplying  it  by  10,  we  have  3  =  30  tenths,  which 
divided  by  8  gives  3  tenths  for  a  quotient.  But  there  are 
6  tenths  remainder.  Reducing  these  to  hundredths,  we  have 
60  hundredths,  which  divided  by  8  gives  7  hundredths  for 
a  quotient.  But  there  are  4  hundredths  remaining.  Re- 
ducing these  to  thousandths,  we  have  40  thousandths,  which 
divided  by  8  gives  5  thousandths  for  a  quotient. 

Thus,    f  —  3   tenths  +  7   hundredths  +  5   thousandths 


96  DECIMAL  FRACTIONS.  [CHAP.  X. 

=0*375.     Hence,  to  reduce  a  common  fraction  to  a  deci- 
mal, we  have  this 

RULE. 

Perform  the  division  expressed  by  the  fraction,  annexing 
as  many  naughts  to  the  numerator  as  are  necessary  to  pro- 
duce a  sufficiently  exact  quotient.  In  the  quotient  point  off 
as  many  decimal  places  as  there  have  been  naughts  annexed. 

NOTE.  —  After  having  annexed  one  0,  if  the  dividend  will  not  con- 
tain the  divisor,  write  0  in  the  quotient,  and  so  on. 

EXAMPLES. 

276-297.  Reduce  to  their  equivalent  decimal  fractions 
the  following  common  fractions  :  ^  ;  J  ;  i  ;  -J  ;  -j-L  ;  |-  ;  J  ; 


298-329.  Reduce  to  decimals  the  following  :  f  ;  f  ;  f-  ; 

3.3.4.4.4.        4.5.5.5.5.        6     .       fi     .       6     . 

T<5"»   T3~>   T»    ¥»   TTMT3"'    T>    ¥  >    9  '   TT  '    TT  »   T3~  »   T7  » 
J  '   T7T  '   T73  '   TT  »   rV  '   TT  '    TT  '>   TT  5   TV  '    TT  >    if  5    if  ' 


330-333.  What  is  the  decimal  value  of  |  of  f  ?  of  J  of 
TT  of  T72  ?  of  f  of  1  divided  by  f  of  |  ?  of  J  of  f  diminish- 
ed by  f  of  I  ? 

334-336.  Find  the  decimal  value  of  T+A+f  +T+TT  » 
of  fxT9TxfxfxT4T;  of  (f+T9T)xf 

337.  A  man  received  7T2j  of  a  dollar  at  one  time,  3j  dol- 
lars at  another,  and  5^  of  a  dollar  at  another.  How  much 
did  he  receive  in  all  ? 

It  will  he  seen,  as  in  some  of  the  preceding  examples,  that  the 
figures  of  the  quotient  are  repeated:  £  giving  0*33  3,  <fcc.,  and  | 
giving  0-1428571428,  <fec.  ;  £  giving  01666,  <fec.  ;  -^  giving  G'08333,  (fee. 


§  64.]  DECIMALS  TO  COMMON  FRACTIONS.  97 

These  are  called  repeating  decimals.  The  figures  that  are  repeat- 
ed are  called  the  repetend,  and  are  distinguished  bj  a  dot  placed  over 
the  first  and  last ;  as  0'333,  &c.=O3  ;  0142857 1428,  <fcc.=od42857  ; 
0-08333,  <fcc.=0-OS3  ;  <fcc. 

When  decimal  figures  precede  the  repetend  they  are  called  the 
finite  part  of  the  decimal.  Thus,  in  0-08333,  <fcc.,  which  is  the  decimal 
value  of  y  £,  0'08  is  the  finite  part. 

REDUCTION  OF  DECIMALS  TO  COMMON  FRACTIONS. 

§  64.  Change  0'375  to  a  common  fraction. 
0'375=T3o7(j5^=f .     Hence,  to  reduce  a  decimal  to  a  com- 
mon fraction,  the  following 

« 

RULE. 

Erase  the  decimal  point ;  supply  the  decimal  denomina- 
tor, and  reduce  the  fraction  to  its  lowest  terms. 

EXAMPLES. 

338-351.  Reduce  to  their  equivalent  common  fractions 
the  following  decimals:  0*5;  O'lo  ;  0'25 ;  0-375;  0'225 ; 
0-435  ;  0-575  ;  0'486  ;  0'656  ;  0'0025  ;  0-00375  ; 
0-000225  ;  O'lOOl  ;  0'36984. 

352-357.  Reduce  to  decimals  the  following:  0*0982; 
0-00764;  0*00025;  0-5005;  0*0125;  0'01250505. 

§  65.  To  reduce  repeating  decimals  to  common  fractions. 

^=0*1111,  <fec. ;  ^J==0'01010101,  <fcc. ;  ^=0-001001001001,  <fcc. ; 

•j=o-3333,  <fcc. ;  5^=0'03030303,  <fcc.  ;  ^=0-003003003003,  <fec. ; 

|=0'6666,  <fec.  Hence  we  see  that  the  numerator  of  the  fraction 
is  the  repeating  decimal  (omitting  the  useless  naught  or  naughts  at 
the  left),  while  the  denominator  of  the  fraction  is  as  many  9's  as  there 
are  figures  in  the  repetend  :  0-0909,  <fec.=^  ;  O'i42857=££f f f£  ; 

I2 

8-1666,  <fcc.=3-16=3-l|=3+-jU=3-|.     Hence  this 

0 


98  DECIMAL  FRACTIONS.  [CHAP.  X. 

RULE. 

Make  the  repetend  the  numerator,  and  as  many  9*5  as  there 
are  figures  in  the  repetend  the  denominator,  of  the  required 
fraction. 

NOTE, — If  there  be  a  finite  part  to  the  decimal,  write  the  whole 
decimal  as  a  complex  fraction  (as  above),  and  reduce  it  to  a  simple 
fraction. 

EXAMPLES. 

358-370.  Reduce  to  common  fractions  the  following  re- 
peating decimals  :  0'3  ;  0'3456  ;  0'4356  ;  0'6543  ;  O'l  ; 

0-12;vO-083=— ^-;     0*06;     0*243;     0*0142857;      0*9  J 

0-0*12345679  ;  0'123321. 

371-385.    Reduce  to  common  fractions  the   following  : 

0-123456;  0*0714285;  0*03571428;  0*027;  0*123; 
0-321;  0-36;  0'63 ;  0*0398;  0*6345;  0*6534;  0*5643; 
0-5634;  0*72;  0*54. 

FEDERAL  MONEY. 

§  66.  Federal  Money  is  the  currency  of  the  United  States. 
This  currency  is  expressed  decimally.  Its  unit  takes  the 
name  of  dollar  ;  its  tenth  is  the  dime  ;  its  hundredth  is  the 
cent;  its  thousandth  is  the  mill.  Thus,  2*345  expresses 
(§52)  2  units,  3  tenths,  4  hundredths,  and  5  thousandths, 
or  2  and  345  thousandths.  Now  writing  the  symbol,  $,* 
before  the  sum,  thus,  $2-345,  it  becomes  at  once  2  dollars, 


*  This  symbol  probably  represents  a  U  placed  upon  an  S  to  denote  the  currency 
of  the  U.  S.  (United  States).  It  must  always  be  written  before  numbers  express- 
ing dollars ;  and  before  numbers  expressing  parts  of  a  dollar,  unless  these  are  de- 
noted by  the  sign  cts.  placed  after  them. 


§  titi.]  FEDERAL  MONEY.  99 

3  dimes,  4  cents,  5  mills  ;  or  2  dollars,  34  cents,  5  mills. 
The  dime  is  always  read  as  cents.  There  is  another  denom- 
ination of  this  currency,  answering  to  the  ten,  called  the 
eagle.  This  is  read  as  dollars.  Thus,  $23  expresses  2  eagles 
and  3  dollars,  but  it  is  read  23  dollars.  $84-6923  is  read 
84  dollars,  69  cents,  2  mills,  and  3  tenths  of  a  mill. 
The  following  is  the  table  of  Federal  Money  : 

10  mills*  (marked  m.)  make  one  cent,  marked  ct. 
10  cents          .         .         "  dime,       "       d. 

10  dimes         .         .         "  dollar,     "       $. 

10  dollars       .         .         "       "   eagle,      "       E. 

The  coins  of  the  United  States  are  the  double-eagle,  engle, 
half-eagle,  quarter- eagle,  dollar,  made  of  gold ;  the  dollar, 
half-dollar,  quarter- dollar,  dime,  half-dime,  and  three  cent 
piece,  made  of  silver ;  the  cent  and  half  cent,  made  of  copper. 

The  mill  is  not  coined. 

NOTE. — The  gold  for  coinage  is  not  pure,  but  consists  of  ££  of  pure 
gold,  -J^-  of  silver,  and  Jj-  of  copper ;  or,  as  usually  expressed,  22  carats 
of  gold,  1  of  Silver,  and  1  of  copper.  A  carat  is  ^  part  of  the  whole. 

The  standard  for  silver  is  1489  of  pure  silver  to  179  of  pure  cop- 
per ;  which,  in  carats,  is  21T5/¥  of  silver,  and  2T8^  °f  copper. 

The  copper  coins  are  of  pure  copper.  The  three  cent  piece  is  |  sil- 
ver and  4  copper. 

By  an  act  of  Congress,  approved  January  18,  1837,  the  gold  and 
silver  coin  must  consist  of  y^o— 77  pure  metal,  and  ^^=-^w 
alloy.  The  alloy  for  silver  must  consist  of  pure  copper,  and  the  alloy 
for  gold,  of  copper  and  silver,  provided  that  the  silver  does  not  ex- 
ceed one-half  of  the  whole  alloy. 

The  weight  of  the  Eagle  was  fixed  at  258  grains  ;  the  weight  of 
the  Dollar  at  412$  grains  ;  that  of  the  Cent  at  168  grains. 


*  The  word  mill  is  from  the  Latin  mille,  meaning  a  thousand ;  cent  from  the 
Latin  centum,  meaning  a  hundred ;  dime  is  from  the  French  word  disme,  mean- 
ing ten. 


100  DECIMAL  FRACTIONS.  [CHAP.  X. 

EXAMPLES. 

386-392.  Read  the  following  :  $7'84  ;  $92'06  ;  $672'123  ; 
$8961-006;  $4180'9673;  $901'001  ;  $3'03. 

393-401.  Read  the  following:  $6'82;  $7'448  ;  $9'02  ; 
$3-01;  $4-07;  $6'93  ;  $48'761  ;  $2l7'001;  $36'987. 

402-408.  Express  in  figures  thirty-seven  cents  ;  forty- 
four  cents,  three  mills  ;  six  dollars,  two  cents  ;  four  dollars, 
eight  mills  ;  nine  dollars,  twenty  cents,  six  mills  ;  five  thou- 
sand dollars,  eight  cents,  nine  mills  ;  one  million  dollars,  one 
and  one-half  cents. 

NOTE.  —  Write  the  half-cent  always  as  5  mills. 

409-412.  Write  37£  cts.  ;  2  dolls.  12^  cts.  ;  4  dolls. 
cts.  ;  5  dolls.  87  cts. 


§  67.  From  the  table  it  is  seen  that  a  number  expressing 
dollars  will  express  cents  by  annexing  two  naughts  to  it,  and 
will  express  mills  by  annexing  three  naughts  ;  thus  3  dolls. 
become  300  cts.  and  3000  mills. 

So  cents  become  mills  by  annexing  one  naught  ;  thus, 
7  cts.  =  70  mills. 

Reversely,  mills  become  cents  fey  cutting  off  a  naught 
from  the  right  ;  and  become  dollars  by  cutting  off  three 
naughts  ;  thus,  8000  mills  =  800  cts^S  dolls. 

Cents  become  dollars  by  cutting  off  two  naughts  from 
the  right  ;  thus,  700  cts.  =  7  dolls. 

NOTE.  —  It  is  obvious  that  this  reduction  of  dollars  to  cents  or  mills 
is  simply  the  multiplication  of  the  sum  expressing  dollars  by  100  or 
1000  ;  and  that  the  reduction  of  cents  or  mills  to  dollars  is  simply 
the  division  of  the  sum  given  by  100  or  1000. 

EXAMPLES. 

413-419.  Reduce  first  to  cents,  then  to  mills,  the  follow- 
ing sums:  $8;  $894;  $620;  $34;  $936273;  $841904; 
$123456. 


§  68.]  FEDERAL  MONEY.  101 

420-426.  Reduce  the  following  sums  to  mills  :  83  cts. 
91cts. ;  4  cts. ;  378  cts. ;  1234  cts. ;  9100  cts.  ;  875618  cts 

427-441.  Reduce  to  dollars  the  following  sums  :  841  cts. 
928  cts.;  4670  cts.;  12986  cts.;  4810  mills;  1234  m. 
4968  cts.  ;  321946m.;  135792  cts.  ;  9800m.;  9800  cts. 
3918762  m. ;  4987621  cts.;  3076009  cts.;  4876543  m. 

442-450.  Reduce  to  cents  the  following  sums  :  8940  m. 
92801  m.  ;  1234567  m. ;  $81*07;  $83*96;  $487'80 
$9654-21;  $13498*20;  $482*31. 

451-456.  Reduce  to  mills  the  following  sums  :  $0*83 
$98-436;  $2'076  ;  $28T296  ;  $4812*37;  $69874*983. 


PROMISCUOUS    EXAMPLES    IN    FEDERAL    MONEY. 

§  68.     The  rules    that  apply  to  operations   in  Decimals 
apply  without  change  to  operations  in  Federal  Money. 

457.  Bought  a  box  of  raisins  for  $1'75,  a  bushel  of  ap- 
ples for  $0-375,  a  cheese  for  $3-l75,  a  barrel  of  sugar  for 
$15'50.     What  did  the  whole  amount  to  ? 

458.  A  farmer  receives  $15'375  for  a  cow,  $75  for  a  horse, 
$3 '125  for  some  potatoes,  $5 '5 5  for  some  poultry.     How 
much  did  he  receive  in  all  ? 

459.  A  person  bought  some  velvet   for  $3*333,   some 
broadcloth  for  $18'75,  some  silk  for  $12*50,  some  cotton 
cloth  for  $5-405,  a  shawl  for  $12*25,  some  carpeting  for 
$30*05.     What  did  the  whole  amount  to  ? 

460.  A  person  borrowed  $2 13 '3 75,  of  which  he  has  paid 
$107-18.     How  much  does  he  still  owe  ? 

461.  Bought  a  cow  for  $13-25,  paid  $6*875.     How  much 
remains  unpaid  ? 

462.  What  will  185  pounds  of  coffee  cost,  at  $0'138  per 
pound  ? 

9* 


102  DECIMAL  FRACTIONS.  [CHAP.  X. 

463.  Bought  8-375   cords  of  wood,  at  $2'50  per  cord. 
What  did  it  cost  ? 

464.  What  will  121 '5  gallons  of  molasses  come  to,  at  41 
cents  per  gallon  ? 

465.  The  length  of   the  Erie    Canal  is  364  miles,   and 
it  cost  $7143790.      What  was  the  average   expense  per 
mile? 

466.  The  Crooked  Lake  Canal  is  8  miles  long,  and  cost 
$156777.     How  much  is  this  per  mile  ? 

467.  In  1842,  the  whole  number  of  children  taught  in  the 
district  schools  of  the  State  of  New  York  was  598901  ; 
the    whole    amount    disbursed  for    common    schools    was 
$1155419'90.     How  much  was  that  per  scholar? 

468.  The  salary  of  the  President  of  the  United  States  is 
$25000.     How  much  is  that  each  day  ? 

469.  In  one  rod  there  are  16 '5  feet.     How  many  rods  in 
3573  feet  ? 

470.  Bought  a  farm  of  137  acres  for  $5324.     How  much 
was  that  per  acre  ? 

471.  If  35  miles  of  railroad  cost  $400000,  how  much  was 
the  average  cost  per  mile  ? 

472.  A  farmer  sells  his  butter  for  $0'21  per  pound,  re- 
ceiving $1613-22.     How  many  pounds  did  he  sell  ? 

473.  The  butter  made  from  the  milk  of  53  cows,  during 
the  summer,  having  been  sold  for  $0'20  per  pound,  brought 
$1579'40.     How  many  pounds  were  sold,  and  what  was  the 
average  produce  of  each  cow  ? 

474.  In  a  dairy  of  46  cows,  suppose  each  averages  2*5 
gallons  of  milk  daily,  and  that  each  gallon  produces   1*1 
pounds  of  cheese,  how  many  pounds  will  be  thus  made  in 
5'7  months  of  30  days  each,  and  what  will  the  whole  bring 
at  15  cents  per  pound  ? 


68.]  FEDERAL  MONEY.  103 

475.  A  farmer  sold  as  follows : 

15127  pounds  of  cheese,  at      6*75  cents  per  pound. 

400        "       "  butter,  "15  "  "  " 

2400        "       "  pork,  "       5  "  "  " 

53  bushels  "  wheat,  "   125  "  "  bushel. 

73        "       "  barley,  "     50  "  "  " 

231         "       "  corn,  "     50  "  "  " 

262        "       "  oats,  "     30  "  "  " 

What  did  the  whole  amount  to  ? 

476.  In  1845,  the  revenue  or  interest  from  the  School 
Fund  of  the  State  of  New  York  was  $86828*96.     During 
the  same  year  there  were  employed  7147  teachers.     If  the 
above  sum  were  equally  divided  among  those  teachers,  what 
would  each  one  receive  ? 

477.  A  compositor  worked  nine  months,  and  during  that 
time  set  up  at  the  rate  of  7000  m's  per  day.     How  many 
thousand  m's  did  he  set  up,  reckoning  25  working  days  to 
the  month?  and  how  much  did  he  receive  at  15  cents  per 
1000  m's? 

478.  A  man,  in  balancing  his  family  accounts  for  one  year, 
found  his  expenses  as  follows  :  for  January,  $9  8 '41  ;    for 
February,  $81*33  ;  for  March,  $102*28  ;  for  April,  $125*26  ; 
for  May,  $74*38  ;  for  June,  $73*47  ;  for  July,  $65*98;  for 
August,    $87'21  ;    for   September,   $70*34 ;    for    October, 
$122*08;  for  November,   $79'68 ;  for  December,   $52*77. 
His  salary  was  $1050  per  annum.     What  had  he  left  at  the 
end  of  the  year  ? 

479.  A  butcher,  a  shoemaker,  and  a  tailor  gave  orders 
on  each  other  in  the  way  of  their  business,  and  at  the  end 
of  a  year  settled  accounts.     The  butcher's  bill  against  the 
tailor  was  $61*84  ;  against  the   shoemaker,  $39*44.     The 
shoemaker's  bill  against  the  butcher  was  $24*30 ;  against 


104:  DECIMAL  FRACTIONS.  [CHAP.  X. 

the  tailor,  819*15.  The  tailor's  bill  against  the  butcher  was 
$42*07 ;  against  the  shoemaker,  $39*39.  Who  received 
balances  in  cash  ? 

480.  Bought  116  feet  of  pine  wood,  at  $4*50  per  cord  of 
128  feet.     How  much  did  I  pay  for  the  load? 

481.  I  bought  19  baskets  of  coal,  at  12  J  cents  per  bush- 
el ;*  lj  cords  of  wood,  at  $8  per  cord ;  3  tons  hard  coal,  at 
$6 '50  per  ton  ;  and  paid  91  cts.  for  sawing  and  splitting  the 
wood.     How  much  did  I  pay  for  my  fuel  ? 

482.  Mr.   Holden's  expenses  for  February  were  as  fol- 
lows :  for  the  table  $28*28  ;  for  sundries  $45*83  ;  for  cloth- 
ing $32*73  ;  oil  $0*68  ;  rent  $14'50  ;  wages  $6*50.     How 
much  in  all  ? 

483.  A  man   takes  50  dollars  to  pay  his  grocer's  bill, 
which  is  as  follows  :  38  doz.  eggs,  at  12^  cts.  a  dozen  ;  34 
pounds  of  white  sugar,  at  11  cts.  a  pound;  42  pounds  of 
brown  sugar,  at  7  cts.  a  pound ;  27  pounds  codfish,  at  3j 
cts.  a  pound ;  3  brooms,  at  1 8  cts.  a  piece ;  7  gallons  ale, 
at  25  cts.  a  gallon  ;  40  pounds  butter,  at  18  cts.  a  pound  ;  2 
galls,  molasses,  at  40  cts.  a  gal. ;  J  gross  matches,  at  62|  cts. 
a  gross.     How  much  change  must  the  man  receive  ? 

484.  The  largest  gold  coin  known  is  the  dobraon  of  Por- 
tugal, of  the  value  of  $32*706.     How  many  double-eagles 
are  there  in  75  dobraons? 

485.  A  piece  of  silk  is  two-thirds  as  wide  as  a  piece  of 
mousseline-de-lain.     It  requires  10  yards  of  the  latter  for  a 
dress.     The  mousseline  is  87-|  cts.  a  yard,  and  the  silk  62^. 
What  is  the  difference  in  price  betweeen  two  dresses  of  equal 
fulness,  one  of  the  silk  and  the  other  of  the  mousseline  ? 

486.  The  receipts  from  United  States  customs  for  tlje  year 
1847-8  were  $31757071 ;  and  from  lands,  &c.,  $3679680. 

*  Each  basket  contains  3  bushels. 


§  68.]  FEDERAL  MOSEY.  105 

The  expenditures  for  the  same  time  were  for  the  army 
$27280163 ;  navy,  $9406737  ;  civil  and  miscellaneous, 
$5585070.  What  was-  the  excess  of  receipts  over  expen- 
ditures ? 

487.  Receipt  the  following  bill  for  its  true  amount. 

HENRY  PHELPS  To  AARON  MUNDIN,        Dr. 

To  75  yds.  Brussels  carpeting,  at  $T42  per  yard; 
"   63  skeins  silk,  at  2£  cts.  a  skein  ; 
"      1  piece  cotton,  31  yds.,  at  11  cts.  a  yard  ; 
"     1  Mousseline  dress,  10  yds.,  at  92  cts.  yd. ; 
"     1  box  hooks  and  eyes,  at  $2'30  ; 
"      1  Piano  cover,  $7'12i;  Table  do.,  $3'25. 

488.  Receipt  the  following  bill  for  its  true  amount. 

JOHN  Cox  To  PHIL.  BRADY,        Dr. 

To  1  sup.  broadcloth  coat,  $22  ; 
"    1  vest,  $5-37  ;  1  pair  pants,  $8'50  ; 
"   Overcoat,  $26  ;  6  pairs  gloves,  at  $O33  per  pair  ; 
"   Suspenders,  $0'50  ;  4  pairs  of  drawers,  £0'S8  per  pair  ; 
"    1  doz.  shirts,  $1-87  each  ;  18  prs.  socks,  at  $0'22  per  pair. 

489.  What  was  the  amount  of  my  butcher's  bill  ?     The 
items  were  as  follow  : 

10  pounds  beef,  at  14  cts.  per  pound  ;  6  pairs  of  fowls,  average  2£ 
pounds  each,  at  18  cts.  per  pound;  kit  mackerel,  25  pounds,  at  5$ 
cts.  a  pound;  38  pounds  sausages,  at  11  cts.  a  pound;  fore-quarter 
lamb,  7  pounds,  at  8  cts.  a  pound  ;  1  bushel  of  potatoes,  75  cts. ;  30 
pounds  lard,  at  10  cts.  a  pound. 

490.  How  many  volumes  of  good  books,  averaging  50 
cts.  a  volume,  could  a  man  purchase  with  the  sum  he  would 
spend  for  rum  (two  glasses  a  day,  at  3  cts.  a  glass),  during 
30  years  of  his  life,  allowing  365  days  to  the  year? 

491.  If  a  man  spend  9  cts.  a  day  for  cigars,  how  much 
will  he  spend  during  a  life  of  70  years,  in  that  worse  than 
useless  indulgence.? 


106  DECIMAL  FRACTIONS.  [CHAP.  X. 

492.  Bought  28J  barrels  of  beef  for  $285,  and  sold  them 
at  a  profit  of  $1*78  per  barrel.  How  much  did  I  sell 
them  for  ? 

§69.  To  find  the  value  of  articles  estimated  by  the  100 
or  1000. 

What  is  the  value  of  9425  bricks,  at  $3'25  per  1000  ? 

Supposing  the  price  to  be  $3'25  for  each  brick,  we 
multiply  the  price  per  brick  by  the  number  of  bricks  ; 
that  is,  $3'25  by  9425  ;  or,  what  is  the  same  thing, 
9425  by  3*25,  the  number  of  dollars,  since  this  is  more 


9425 
3-2o 

47125 
18850 


convenient.  28215 

The  product  30631'25  is  evidently  1000  times  too 
great.  We  therefore  divide  it  by  1000  (§  62),  by  re-  30631-25 
moving  the  decimal  3  places  to  the  left.  The  true 
result,  then,  is  $30-63125.  Had  the  bricks  been  $3'25  per  100,  the 
decimal  should  have  been  removed  two  places  to  the  left.  Hence 
this 

RULE. 

Multiply  the  number  of  articles  by  the  number  expressing 
the  price  per  100  or  1000.  From  the  right  of  the  product 
point  off  two  figures  when  the  articles  are  estimated  by  the  100, 
or  three  figures  when  they  are  estimated  by  the  1000. 

NOTE. — The  decimal  figures  pointed  off  by  this  rule  are  in  addition 
to  those  which  are  pointed  off  by  the  usual  rule  for  multiplication 
of  decimals. 

EXAMPLES. 

493.  What  is  the  value  of  1300  feet  of  hemlock  boards, 
at  $5-50  per  1000  ? 

494.  What  is  the  value  of  6Yo  feet  of  clear  pine  stuff,  at 
$25  per  1000  ? 

495.  What  is  the  value  of  11035  feet  of  timber,  at  $2'25 
per  100  ? 


§  70.]  FEDERAL  MONEY.  107 

496.  What  is  the  value  of  90422  bricks,  at  $3'75  per 
1000? 

497.  What  must  be  paid  for  laying  875  bricks,  at  $3*25 
per  1000? 

498.  What  cost  1689216  laths,  at  8  cts.  per  100? 

499.  A  man  carted  575  loads  of  bricks,  each  load  con- 
taining 1800  bricks.     What  cost  the  whole,  at  $8 '25  per 
1000? 

500.  What  must  be  paid  for  planing  4976280  feet  of 
boards,  at  42  cts.  per  1000  feet? 


AN  ABRIDGED  METHOD  FOR  OPERATIONS  IN  FEDERAL  MONEY, 

by  the  aid  of  aliquot  parts.     (See  §  116.) 

§  70.  What  cost  704  yards  of  cloth,  at  12^  cts.  per  yard  ? 

The  question  may  be  answered  in  the  usual  way  by  multiplying 
704  by  0-125,  the  cost  of  one  yard  in  dollars.  But  there  is  a  shorter 
method. 

If  the  price  of  the  cloth  had  been  1  dollar  a  yard,  the  704  yards 
would  have  cost  704  dollars.  But  12^  cts.  is  |  of  a  dollar  ;  conse- 
quently, the  704  yards  must  cost  1$±  dollars=f  88. 

At  12^  cts.  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  $88  ? 

This  question  might  be  answered  in  the  usual  way  by  dividing  88 
by  0-125.  But  as  12£  cts.  is  |  of  a  dollar,  1  dollar  would  buy  as 
many  yards  as  £  is  contained  times  in  1 ;  that  is,  l-)-i=8,  and  88 
dollars  would  buy  88  times  as  many  yards ;  that  is,  88-r|=704,  the 
number  of  yards.  Hence  this 

RULE. 

As  the  conditions  of  the  question  require  division  or  mul- 
tiplication, divide  or  multiply  by  the  fractional  part  of  the 
dollar  which  the  price  expresses. 


108  DECIMAL  FRACTIONS.  [CHAP.  X. 

TABLE 

Of  fractional  parts  of  a  dollar,  called  aliquot  or  exact  parts. 

cts.       $  cts.        $  eta.        $  cts.      $ 

5  =&.  16|=  i.  33J=  i.  62i=f. 

6J=A-  18f=A-  371=  a.  66§=f. 

8J=A.  20  =  J.  50  =  A  75  =J. 

10  =TV  25  =  ]>  66J=TV  83j=f 

121=   ..  3U=#.  58J=&.  871=}. 

NOTE.— 121  cts.  would,  by  the  table,  be  £  of  a  dollar;  $1-1 2$ 
would  be  §  ;  $2-31*  would  be  $aj  ;  $4-75  would  be  §-'/-,  <fca 

EXAMPLES. 

501-514.  What  would  678  baskets  of  peaches  cost,  at 
12£  cts.  a  basket  ?  at  16f  cts.  ?  at  18f  cts.  ?  at  20  cts.  ?  at 
25  cts.  ?  at  31-i  cts.  ?  at  33j  cts.  ?  at  37J  cts.  ?  at  50  cts.  ? 
at  62£  cts.  ?  at  66f  cts.  ?  at  75  cts.  ?  at  83j  cts.  ?  at  87£ 
cts.  ? 

515-521.  What  would  840  yards  of  cloth  cost,  at  one 
shilling,  New  York  currency  (12j  cts.),  per  yard  ?  at  3  shil- 
lings ?  at  4  ?  at  5  ?  at  6  ?  at  7  ?  at  2  ? 

NOTE. — It  would  be  more  simple  and  more  consistent  that  all  ac- 
counts in  the  United  States  should  be  kept  in  Federal  Money.  Yet 
in  most  of  the  States  the  old  colonial  denominations  of  shillings  and 
pence  are  more  or  less  used.  The  unit  of  these  denominations  is  not 
of  the  same  value  in  all  the  States  ;  thus,  in  New  York,  1  shilling= 
12£  cts. ;  in  New  England =16  §  cts.  The  reason  of  this  is,  that  at  the 
time  of  the  adoption  of  Federal  Money  in  1786,  the  paper  currency 
of  the  several  colonies  had  depreciated  in  value.  A  colonial  pound 
or  shilling  was  not  worth  so  much  as  a  sterling  (English)  pound  or 
shilling.  But  this  depreciation  being  unequal  in  the  several  colonies* 
their  shillings  and  pence  varied  in  value.  This  variation  continues 
to  this  day. 

522-527.  At  one  shilling,  N.  E.  currency  (16 J  cts.),  per 
pound,  how  many  pounds  of  butter  can  be  bought  for  $0  ? 


§  71.]  DENOMINATE  NUMBitt&.  109 

for  $12?  for  $54?  for  $926?  for  8217  dolls.?  for  98127 
dolls.  ? 

528-538.  How  many  pecks  of  apples  will  $37  buy,  at 
fourpence-ha'penny,  N.  E.  currency  (6j  cts.),  a  peck  ?  at 
ninepence  (12J  cts.)  a  peck?  at  one  shilling  a  peck?  at 
sixpence  (8§-  cts.)  a  peck?  at  4  and  6  pence  a  peck?  at  2 
and  6  pence  ?  at  3  and  6  pence  ?  at  5  and  6  pence  ?  at  8 
shillings  ?  at  4  shillings  ?  at  5  shillings  ? 

539-549.  How  many  brushes  will  $50  purchase  at  a  six- 
pence, New  York  currency  (6j  cts.),  a  piece  ?  at  a  shilling 
a  piece?  at  18  pence  (18$  cts.)  apiece  ?  at  2  shillings  ?  at 
2  and  6  pence  ?  at  3  shillings  ?  at  3  and  6  pence  ?  at  4  shil- 
lings ?  at  5  ?  at  6  ?  at  7  ? 

550-562.  For  656  dollars,  how  many  yards  of  carpeting 
can  I  buy,  at  37^  cts.  per  yard  ?  at  50  cts.  ?  at  56^  cts.  ? 
at  58J  cts.  ?  at  62|  cts.  ?  at  66f  cts.  ?  at  75  cts.  ?  at  83j 
cts.?  at$l-12i?at$l'25?  at$l'33-J-?  at$l'50?  at$l'62£? 

563-575.  What  will  98  yards  of  carpeting  cost,  at 
cts.^eryard?  at$l-05?  at$l-06i?  at$l'08£? 
at$l'16f  ?  at  $1-25?  at$l'33j?  at$l'62£?  at 
at$l-56i?  at  $1-50?  at$l'87£? 


CHAPTER   XI. 

DENOMINATE   NUMBERS. 

§71.  WE  have  thus  far  treated  of  abstract  numbers; 
that  is,  of  numbers  as  simple  units,  or  as  parts  of  a  unit. 
We  shall,  in  the  present  chapter,  treat  of  denominate  num- 
bers, or  numbers  having  reference  to  particular  things. 

(MO 

10 


110  DENOMINATE  NUMBERS.  [CHAP.  XI. 


.  —  Federal  Money,  when  considered  as  units,  tenths,  hun- 
dredths,  &c.,  has  an  abstract  character  ;  when  considered  as  dollars, 
cents,  and  mills,  has  a  denominate  character. 

In  multiplication,  the  multiplicand  being  repeated  a  certain  num- 
ber of  times,  or  a  certain  fraction  of  a  time  when  the  multiplier  is  a 
fraction,  it  follows  that  the  multiplier,  considered  as  a  multiplier, 
must  always  be  regarded  as  an  abstract  number.  And  since  the 
product  is  a  repetition  of  the  multiplicand,  it  must  be  like  the  mul- 
tiplicand ;  that  is,  if  the  multiplicand  is  an  abstract  number,  the 
product  must  be  an  abstract  number  ;  if  the  multiplicand  is  a  de- 
nominate number,  the  product  must  be  a  denominate  number  of  the 
same  kind. 

In  division,  when  the  quotient  shows  how  many  times  the  divisor 
is  contained  in  the  dividend,  or  what  fraction  of  a  time  when  the  di- 
visor is  greater  than  the  dividend,  it  follows  that  the  quotient  must 
be  regarded  as  an  abstract  number,  and  that  the  divisor  and  divi- 
dend must  be  alike. 

When,  however,  the  process  of  division  is  rather  the  di- 
viding of  a  dividend  into  as  many  equal  parts  as  are  indi- 
cated by  the  divisor,  the  quotient  expressing  the  units  in  one 
of  those  parts  is  of  the  same  kind  as  the  dividend,  while 
the  divisor  is  to  be  regarded  as  an  abstract  number.  See 
Example,  §  91. 

The  following  are  the  most  important  tables  of  weights 
and  measures,  &c.,  and  must  be  thoroughly  learned  by  the 
pupil. 

§  72.  ENGLISH  OR  STERLING  MONEY;. 

4  farthings  (qr.  or  far.)  make  1  penny,     d. 
12  pence  "      1  shilling,   s. 

20  shillings  "      1  pound,     £ 

NOTE  1.  —  Farthings  are  often  expressed  as  fractions  of  a  penny 
Thus,  Ifar.—^d.  ;  Ifar.—^d,;  3/ar.=f<£ 

NOTE  2.  —  The  pound  sterling  was  originally  a  bank  note  ;  but  the 


§  73,  74.]  DENOMINATE  NUMBERS.  Ill 

note  has  fallen  into  disuse  ;  and  a  gold  coin,  called  a  sovereign,  of 
the  value  of  $4-84,  supplies  its  place. 

NOTE  3. — The  symbol  £  is  used  because  it  is  the  first  letter  of  the 
Latin  word  libra,  which  signifies  a  pound  ;  s.  stands  for  solidus,  which 
signifies  a  shilling  ;  d.  for  denarius,  a  penny  ;  g.  for  quadrans,  a  quarter. 

The  stroke /of ten  written  between  shillings  and  pence  is  a  corrup- 
tion of  the  long/. 

§73.     TROY  WEIGHT. 

24  grains  (gr.)   make  1  pennyweight,    pwt. 
20  pennyweights  "      1  ounce,  oz. 

12  ounces  "      1  pound;  Ib. 

NOTE  1. — The  original  of  all  weights  used  in  England  was  a  grain  cf 
wheat,  gathered  out  of  the  middle  of  the  ear  ;  32  of  these,  well  dried, 
were  to  make  one  pennyweight.  But  at  a  later  period,  it  was 
thought  sufficient  to  divide  the  same  pennyweight  into  24  equal 
parts,  still  called  grains,  being  the  least  weight  now  in  common  use. 

Coins,  precious  metals,  jewels,  and  liquors,  are  weighed  by  Troy 
weight. 

NOTE  2. — This  scale  of  weights  is  said  to  have  been  borrowed  from 
Troyes  in  France — hence  its  name.  Some,  however,  contend  that  the 
name  has  reference  to  the  monkish  title  given  to  London  of  Troy 
Novant. 

§  74.     APOTHECARIES'  WEIGHT. 

20  grains  (gr.)  make  1  scruple,  3 

3  scruples  "  1  dram,  3 

8  drams  "  1  ounce,  % 

12  ounces  "     1  pound,  ftr 

NOTE. — This  weight,  as  its  name  would  imply,  is  used  in  weighing 
medicines  in  small  quantities,  as  for  prescriptions.  But  drugs  and 
medicines  in  gross  are  bought  and  sold  by  Avoirdupois  Weight.  The 
pound  and  ounce  Apothecaries'  Weight  are  the  same  as  in  Troy 
Weight. 


112  DENOMINATE  NUMBERS.  [CHAP.  XI. 

§  75.     AVOIRDUPOIS  WEIGHT. 

16  drams  (dr.)  make  1  ounce,  -  oz. 

16  ounces  "     1  pound,  Ib. 

25  pounds  "     1  quarter,  qr. 

4  quarters  "     1  hundred  weight,  cwt. 

20  hundred  weight       "     1  ton,  T. 

NOTE  1. — By  this  weight  are  weighed  all  things  of  a/  coarse  or 
drossy  nature,  as  bread,  butter,  cheese,  flesh,  groceries,  and  some 
liquids  ;  all  metals,  except  gold  and  silver. 

NOTE  2. — Formerly  28  pounds  were  estimated  as  1  qr.,  112  pounds 
1  cwt.,  and  2240  Ibs.  1  ton.  These  weights  are  still  used  for  cheap 
and  heavy  articles,  such  as  iron,  coal,  plaster,  <fcc. 

NOTE  3. — The  pound  Avoirdupois  contains  7000  grains  Troy,  while 
the  Troy  pound  contains  only  5760  grains. 


§  76.     LONG  MEASURE. 

12     inches  (in.)  make  1  foot,  ft. 

3     feet  "  1  yard,  yd. 

5l  yards  "  1  rod,  perch,  or  pole,  rd. 

40  rods  "  1  furlong,  fur. 

8     furlongs  "  1  mile,  mi. 

3  miles  "  1  league,  L. 

69 \  miles,  nearly,  "  1  degree,  deg.  or  °. 

NOTE  1. — 4  inches  make  1  hand ;  9  inches,  1  span ;  18  inches,  1  cu- 
bit ;  6  feet,  1  fathom  ;  3  feet,  1  pace. 

NOTE  2. — The  inch  is  subdivided  sometimes  into  tenths  ;  sometimes 
into  halves,  quarters,  eighths,  sixteenths ;  and  sometimes  into 
twelfths,  called  lines  or  primes. 

NOTE  3. — A  nautical  or  geographical  mile  is  -J-Q  of  a  degree  of  the 
earth's  circumference.  And  since  one  degree  is  69^  statute  or  legal 
miles,  we  have  1  nautical  mile  equal  to  11^=1'1527  statute  miles 
s=6086§  feet. 


§  77.]  DENOMINATE  NUMBERS.  113 

A  knot,  in  nautical  language,  is  a  division  of  the  log-line  of  jiy 
of  a  nautical  mile.  A  half-minute  glass  is  used  in  connection  with 
the  log,  by  observing  how  many  knots  of  the  log -line  are  run,  while 
the  sand  is  running  from  the  glass.  As  half  a  minute  is  —^  of  an 
hour,  it  follows  that  the  number  of  knots  thus  run  will  be  the  num- 
ber of  miles  the  ship  is  making  hourly.  Hence,  it  is  frequently  said 
that  a  ship  was  running  at  the  rate  of  a  certain  number  of  knots,  by 
which  is  meant  the  number  of  nautical  miles  she  was  making  hourly. 
In  this  sense,  knot  is  used  for  a  nautical  mile. 

NOTE  4. — The  standard  length  of  the  yard  in  the  United  States, 
from  which  all  other  measures  of  length  are  derived,  is  the  same  as 
that  of  the  Imperial  yard  of  Great  Britain.  This  yard  is  deduced 
from  that  of  a  pendulum  which  vibrates  once  in  a  second  in  vacuum 
at  the  level  of  the  sea  at  London.  Such  a  pendulum  is  found  to  be 
39-13929  inches. 

NOTE  5. — The  French  government  derive  their  linear  unit  of  meas- 
ure from  one  quarter  of  the  circumference  of  a  great  circle  of  the 
earth  passing  through  the  poles.  Having  determined  by  actual  sur- 
veys the  length  of  that  portion  of  a  quarter  circle,  which  is  comprised 
between  the  parallels  of  Dunkirk  and  Barcelona,  they  deduced  the 
length  of  the  entire  quarter  from  the  equator  to  the  pole,  and  took 
one  ten-millionth  part  of  it  for  a  metre.  This  method  gave  for  the 
French  metre  39-37079  English  or  United  States  inches,  equal  3 '2809 
feet,  nearly. 

§  77.     CLOTH  MEASURE. 

2j  inches  (in.)  make  1  nail,  na. 

4  nails  "  1  quarter  of  a  yard,  qr. 

3  quarters  "  1  Ell  Flemish,  E.  Fl. 

4  quarters  "  1  yard,  yd. 

4  qr.  1 J  in.         "     1  Ell  Scotch,  E.  S. 

5  quarters  "     1  Ell  English,  E.  E. 

6  quarters  "     1  Ell  French,  E.  Fr. 

10* 


DENOMINATE  MTIMBERS.  [CHAP.  XI. 


§  78.     SQUARE  MEASURE. 

144     square  inches  (sq.  in.)   make  1  square  foot,        sq.ft. 

9     square  feet  "     1  square  yard,      sq.  yd. 

30  J  square  yards  "     1  square  rod  or  pole,  P. 

40     square  rods 

4     roods 
640 


acres 


1  rood, 
1  acre, 
1  square  mil< 


R. 
A. 
M. 


1  foot=12  inches. 


NOTE  1. — This  measure  is  used  for  measuring  surfaces  such  as 
boards,  glass,  pavements,  plastering,  flooring,  painting,  and  any  kind 
of  materiel  or  work  where  length  and  breadth  only  are  concerned. 
It  is  always  employed  for  measuring  land,  and  for  this  reason  is 
sometimes  called  Land  Measure. 

A  square  is  a  figure  having  four 
equal  sides,  and  all  its  angles  right 
angles ;  that  is,  the  sides  are  per- 
pendicular to  each  other. 

If  the  length  of  one  of  the  sides 
is  one  inch,  it  is  called  a  square 
inch;  if  the  length  of  one  of  the 
sides  is  one  foot,  or  12  inches,  it  is 
called  a  square  foot,  which  by  the 
figure  we  see  -is  composed  of  12  X 
12=144  square  inches. 


In  a  similar  manner,  if  we  had  a  square,  each  of  whose  sides  was 

3  feet,  it  would  contain  3X3=9  sq.feet,  or  one  yard. 

NOTE  2. — The  acre  is  always  applied  to  surface  or  area.  There  is 
no  such  thing  as  an  acre  long.  It  is  of  such  a  magnitude  as  not  to 
admit  of  being  accurately  given  in  the  form  of  a  square.  The  same 
is  true  of  the  rood. 

NOTE  3. — In  measuring  land,  Gunter's  chain  is  used  ;  its  length  is 

4  rods,  or  66  feet.     It  is  divided  into  100  links. 


7y^  inches  make 

100  links,  or  4  rods,  or  66  feet,     " 

80  chains  " 

iOOOO  square  links  " 

10  square  chains  " 


1  link,  /. 

1  chain,  c. 

1  mile,  mi. 

1  square  chain,  sq.  c. 

I  acre,  A. 


§  79.]                         DENOMINATE  NUMBERS.  115 

§  79.     SOLID  OR  CUBIC  MEASURE. 

1728  solid  inches    (S.  in.)         make  1  solid  foot,  S.  ft. 

27  solid  feet                                "     1  solid  yard,  S.yd. 

40  feet  of  round  timber  or  )       „     ..  ,  rp 
50  feet  of  hewn  timber       j 
128  solid  feet                                "     1  cord  of  wood,  C. 


1.  —  This  measure  is  used  in  measuring  solid  bodies  or  spaces  ; 
that  is,  things  having  length,  breadth,  and  height  or  thickness  :  such 
as  earth,  stone,  timber,  bales  of  goods,  the  capacity  of  rooms,  <fcc. 

NOTE  2.  —  A  cube  is  a  solid  bounded  3  feet> 

by  six  equal  squares,  resembling  a 
common  tea-chest. 

If  the  sides  of  a  cube  are  each  one 
inch  long,  it  is  called  a  cubic  inch.  If 
each  side  is  one  foot  long,  it  is  called 
a  cubic  foot,  <fec. 

The  figure  represents  a  cut^e,  each 
side  of  which  is  3  feet  or  one  yard  in 
length  ;  consequently,  it  represents  one  solid  or  cubic  yard. 

The  top,  which  is  equal  to  the  base,  contains  3  X3=9  square  feet  ; 
hence,  if  this  was  only  one  foot  in  height,  it  would  contain  9  cubic 
feet  ;  but  as  it  is  3  feet  in  height,  it  must  contain  3  times  9=27  cu- 
bic feet.  Hence,  one  cubic  yard  is  equivalent  to  3X3X3=27  cubic 
feet. 

In  the  same  way  one  cubic  foot  is  equivalent  to  12X12X12=1728 
cubic  inches. 

NOTE  3.  —  A  ton  of  round  timber  is  such  a  quantity  of  timber  in 
its  natural  state  as,  when  hewed,  will  make  40  cubic  feet. 

NOTE  4.—  A  pile  of  wood  4  /V.  wide,  4ft.  high,  and  8ft.  long,  makes 
a  cord.  One  foot  in  length  of  such  a  pile  is  sometimes  called  a  cord 
foot.  It  contains  16  solid  feet:  consequently,  8  cord  feet  make  1 
cord. 


116 


DENOMINATE   NUMBERS.  [CHAP.  XL 


§  80.     WINE  MEASURE. 


4  gills  (ffi.)  make  1  pint, 

2  pints 

4  quarts 

31^  gallons 

63  gallons 

2  hogsheads 

2  pipes 

NOTE. — The  wine  gallon  contains  231  cubic  inches.     The  Imperial 
gallon  of  Great  Britain  contains  277'274  cubic  inches. 


pt. 

I  quart,          qt. 
1  gallon,        gal. 
1  barrel,        bar 
1  hogshead,  hhd. 
1  pipe,          pi. 
1  tun,  tun. 


§  81.     ALE  OR  BEER  MEASURE. 

2     pints  (pt.)  make  1  quart,  qt. 

4     quarts  "     1  gallon,  gal. 

36     gallons          "     1  barrel,  bar. 

1J  barrels          "     1  hogshead,  hhd. 

NOTE. — The  beer  gallon  contains  282  cubic  inches.     Milk  is,  or 
should  be,  measured  by  this  measure. 


§  82.     DRY  MEASURE. 

2  pints  (pt.)  make  1  quart,  qt. 

8  quarts  "     1  peck,  pk. 

4  pecks  "     1  bushel,  bu. 

32  bushels  "     1  chaldron,  ch. 

NOTE  1. — By  this  are  measured  all  dry  wares ;  as  grain,  seeds, 
roots,  fruits,  salt,  coal,  sand,  oysters,  <fec. 

NOTE  2. — The  standard  unit  of  dry  measure  adopted  by  the  Uni- 
ted States  is  the  Winchester  bushel.  This  is  made,  by  English  stat- 
ute, to  contain  2150^  cubic  inches.  It  is  a  measure  of  cylindric 
form,  8  inches  deep  and  18  J  inches  in  diameter. 


§  83.]  DENOMINATE  NUMBERS.  117 


§  83.     TIME. 

60  seconds  (sec.)  make  1  minute,  tnin. 

60  minutes  "     1  hour,  hr. 

24  hours  "     1  day,  da. 

7  days  "     1  week,  wTc. 

4  weeks  "     1  month,  mo. 

13  mo.,  1  da.,  6  hr.,  or  )  *  V  « 

v     "     1  J  ulian  year,  yr. 
365  da.,  6  7w.  J 

NOTE  1. — The  true  length  of  the  solar  year  is  365-242217  days,  or 
about  365  da.  5  hr.  48  tnin.  47-J-  sec. 

NOTE  2. — As  the  year  exceeds  365  days  by  very  nearly  6  hours, 
at  the  end  of  every  4  years  an  additional  day  is  given  to  the  month 
of  February.  The  years  containing  this  extra  day  are  called  Bissex- 
tile or  Leap  Years.  But  since  the  excess  of  which  we  speak  is  not 
quite  6  hours,  the  addition  of  the  extra  day  will  in  time  give  too 
many  days  to  the  calendar  ;  therefore  every  closing  year  of  a  cen- 
tury (called  a  centennial  year)  that  is  not  divisible  by  400  is  regarded 
as  a  common  year. 

Every  year  (except  a  centennial)  that  may  be  divided  by  4  is  a 
Leap  year,  and  has  366  days.  Thus,  1840,  1844,  1848,  were  leap 
years,  as  1852,  1856,  cfec.,  will  be  ;  1800  not  being  divisible  by  400 
was  a  common  year,  but  the  year  2000  will  be  a  leap  year. 

NOTE  3. — In  business  transactions  30  days  are  considered  a  month, 
and  12  months  a  year. 

The  following  lines  will  help  the  pupil  to  remember  the  number 
of  days  in  each  month : 

Thirty  days  hath  September, 
April,  June,  and  November  ; 
All  the  rest  have  thirty-one, 
Excepting  February  alone  : 
To  which  we  twenty-eight  assign, 
Till  leap  year  gives  it  twenty-nine. 

NOTE  4. — It  is  very  desirable  to  be  able  readily  to  determine  the 
number  of  days  from  any  particular  date  to  any  other  date.  For 
this  purpose,  we  will  give  the  following 


118 


DENOMINATE  NUMBERS. 


[CHAP.  xi. 


TABLE, 

SHOWING  THE  NUMBER  OF  DAYS  FROM  ANY  DAY  OF  ONE  MONTH  TO  THK  SAME  DAY 
OF  ANY  OTHER  MONTH  IN  THE  SAME  YEAR. 


FROM  ANY 
DAY  OF 

JANUARY  .... 
FEBRUARY... 

TO  THE  SAME  DAY  OF 

Jun. 

Feb. 

Mar. 

Ap'l 

May. 

June 

July. 

Aug.  Sept. 

Oct. 

.Vov. 

Dec. 

365 
334 
306 
275 
245 
214 
184 
153 
122 
92 
61 
31 

31 
365 
337 
306 
276 
245 
215 
184 
153 
123 
92 
62 

59 
28 
365 
334 
304 
273 
243 
212 
181 
151 
120 
90 

90 
59 
31 
365 
335 
304 
274 
243 
212 
182 
151 
121 

120 
89 
61 
30 
365 
334 
304 
273 
242 
212 
181 
151 

151 
120 
92 
61 
31 
365 
335 
304 
273 
243 
212 
182 

181 
150 
122 
9J 
61 
30 
365 
334 
303 
273 
242 
212 

212 
181 
153 
122 
92 
61 
31 
365 
334 
304 
273 
243 

243 
212 
184 
153 
123 
92 
62 
31 
365 
335 
304 
274 

273 
242 
214 
183 
153 
122 
92 
61 
30 
365 
334 
304 

304 
273 
245 
214 
184 
153 
123 
92 
61 
31 
365 
335 

334 
303 
275 
244 
214 
183 
153 
122 
91 
61 
30 
365 

MAY   

AUGUST  •-. 
SEPTEMBER 
OCTOBER.. 
NOVEMBER 
DECEMBER 

As  an  example,  suppose  we  wish  the  number  of  days  from  No- 
vember 6th  to  the  15th  of  next  April.  We  find  November  in  the 
left-hand  vertical  column,  and  April  at  the  top  line  of  the  table,  and 
at  the  intersection  we  find  151  days.  So  that  from  November  6th 
to  April  6th  is  151  days  ;  consequently,  adding  9,  we  find  160  for  the 
number  of  days  between  November  6th  and  April  15th. 

This  table  is  constructed  on  the  supposition  of  28  days  to  Febru- 
ary. When  there  are  29  days  in  February  the  proper  allowance 
must  be  made. 


84.     CIRCULAR  MEASURE. 


make  1  minute,  ' 

"     1  degree,  ° 

"     1  sign,  s. 

"     I  circle,  cr. 


60  seconds  (") 
60  minutes 
30  degrees 
12  signs  or  360° 

NOTE  1. — By  this  measure  latitude  and  longitude,  and  the  motions 
of  the  heavenly  bodies,  which  appear  to  move  in  circles,  are  esti- 
mated. 

NOTE  2. — Every  circle,  whether  great  or  small,  is  supposed  to  be 
divided  into  360  equal  parts,  called  degrees. 


§  85.]  DENOMINATE  NUMBERS.  119 

NOTE  3. — The  sun  appears  to  pass  completely  around  the  earth  in 
24  hours  ;  that  is,  it  appears  to  move  westward  over  an  entire  circle 
or  360°  of  longitude  in  24  hours.  Consequently,  in  one  hour  it  will 
move  over  ^  of  360°=15°  of  longitude.  Hence,  if  the  difference  in 
the  longitudes  of  two  places  is  15°,  it  will  be  noon  at  the  more  east- 
erly place,  just  one  hour  before  it  is  noon  at  the  other  place.  And 
in  all  cases,  the  difference  in  time  of  any  two  places  will  be  at  the 
rate  of  one  hour  for  every  15°  of  longitude  between  the  two  places. 
As  an  example,  suppose  the  city  of  Washington  to  be  77°  west  of 
Greenwich :  it  is  required  to  find  what  time  it  is  at  Washington, 
when  it  is  noon  at  Greenwich. 

Dividing  77°  by  15°,  we  have  5T2^  for  the  number  of  hours  differ- 
ence in  tune  ;  that  is,  5h.  8m.  And  as  the  apparent  motion  of  the 
sun  is  westward,  it  must  be  earlier  at  Washington  than  at  Green- 
wich. Therefore,  when  it  is  noon  at  Greenwich,  it  is  5h.  8m.  before 
noon  at  Washington  ;  that  is,  6h.  52m.  A.  M. 

§  85.  Measures,  &c.,  not  included  in  the  foregoing  tables. 

12  individual  things  make  1  dozen. 

12  dozens  or  144  "1  gross. 

12  gross  "     1  great  gross. 

20  individual  things  "     1  score. 

112  pounds  "     1  quintal  of  fish, 

196       "  "     1  barrel  of  flour. 

200       "  "     1  barrel  of  pork  or  beef. 

24  sheets  of  paper  "     1  quire. 

20  quires  "     1  ream. 
A  sheet  folded  in    2  leaves  makes  a  folio. 

"     "          "      "     4  "         "a  quarto  or  4to. 

"     "          "      "     8  "         "       an  octavo  or  8vo. 

"     "          "      "12  "         "       a  duodecimo  or  12mo, 

"      «          "      "  18  "         "       an  18mo. 


120  DENOMINATE  NUMBERS.  [CHAP.  XI 

EXERCISES  ON  THE  TABLES. 

1-10.  How  many  farthings  in  2  pence?  in  4?  in  5  ?  in 
8?  10?  15?  20?  25?  50?  100? 

11-20.  How  many  farthings  in  1  shilling  ?  in  2  shillings  ? 
in  3?  5?  8?  15?  20?  25?  50?  100? 

21-30.  How  many  pence  in  2  shillings?  in  3?  5?  7? 
9?  15?  20?  25?  50?  100? 

31-40.  How  many  shillings  in  2  pounds  ?  in  3?  5?  7? 
9?  15?  21?  25?  50  ?  100? 

41-42.  How  many  pence  in  3qr.-}-2qr.-}-6qr.-+-1qr.-{- 


43.  In   a   sovereign   how   many  shillings  ?    how   many 
pence  ?  how  many  farthings  ? 

44.  How  many  farthings  in  2/6  -f  4/9  +  35.  +  6qr.  ? 
45-60.  How  many  grains  in   Zpwt.t  in  5?  in  7?  15? 

25?  50?  How  many  in  1  02.  ?  202.  ?  in  5?  7?  15?  25? 
50?  in  1/6.?  2/65.?  5/65.? 

61-74.  How  many  pennyweights  in  202.  ?  in  5?  7?  9? 
15  ?  20  ?  25  ?  50  ?  in  1/6.  ?  in  3/6s.  ?  in  5  ?  9  ?  15  ?  25  ? 

75-81.  How  many  ounces  in  2/6$.  ?  in  4/6s.  ?  in  9  ?  in 
15?  in  25?  in  50?  in  100? 

82-101.  How  many  grains  in  33  ?  in  5  ?  7  ?  12  ?  18  ?  in 
13?  in  5?  7?  12?  18?  in  15?  in  5?  7?  12?  18?  in 
lib?  in  5?  7?  12?  18? 

102-110.  How  many  scruples  in  23  ?  in  5  ?  in  9  ?  in  15  ? 
in  5  ?  in  9  ?  in  lib  ?  in  5  ?  in  9  ? 

111-118.  How  many  drams  in  2!  ?  in  5  ?  in  7  ?  in  12  ? 
in6ib?  in  9  ?  in  15  ?  in  16? 

119-128.  How  many  drams  avoirdupois  in  2oz.  ?  in  7  ? 
in  9  ?  in  1/6.  ?  in  2  ?  in  9  ?  in  20  ?  in  Iqr.  ?  in  Icwt.  ?  in  1  T.  ? 

129-136.  How  many  ounces  avoirdupois  in  2  /6s.  ?  in  7  ? 
in  9  ?  in  15  ?  in  25  ?  in  100  ?  in  Icwt.  ?  in  IT.  ? 


§  85.]  DENOMINATE  NUMBERS.  121 

137-141.  How  many  pounds  in  IT.  ?  in  3?  n  20  ?  in 
50?  in  100? 

142-150.  How  many  inches  in  2ft.  ?  in  7ft.  ?  in  20  ?  in 
lye?.?  in  5 \yds.  ?  in  1  rod?  in  1  furlong?  in  1  mile?  in 
1  degree? 

151-153.  How  many  feet  in  100  yds.  ?  in  1  mile  ?  in  100  ? 

154-158.  How  many  inches  in  18  hands  ?  in  7  spans  ? 
in  20  cubits  ?  in  1  fathom  ?  in  40  paces  ? 

159-164.  How  many  inches  in  16  nails?  in  1  qr.  ?  in 
1  E.  FL  ?  \E.SA  1  E.  E.  ?  1  E.  Fr.  ? 

165-167.  How  many  sq.  in.  in  42  sq.ft.  ?  in  1  rood?  in 
1  acre? 

168-169.  How  many  sq.  yds.  in  an  acre  ?  in  1  sq.  mi.  ? 

170-172.  How  many  sq.  in.  in  1  sq.  yd.  ?  in  1  T.  hewn 
timber  ?  in  1  C.  ? 

173-177.  In  6  tuns,  wine  measure,  how  many  hhd.  ?  how 
many  bar.  ?  how  many  gal.  ?  how  many  qt.  ?  how  many  pt.  ? 

178-180.  How  many  pts.  in  1  Me?.?  how  many  qts.  ? 
how  many  gi.  ? 

181-182.  How  many  more  pints  in  1  bar.,  beer  measure, 
than  in  1  bar.,  wine  measure  ?  in  1  hogshead  wine  than  in 
1  hhd.  beer  ? 

183-186.  How  many  quarts  in  1  bu.  ?  in  1  ch.  ?  in  15  ch.  ? 
in  25? 

187-189.  How  many  pecks  in  50  bu.  ?  in  50  cA.?  in 
100  ch.% 

190-197.  How  many  seconds  in  5  mm.  ?  in  15  min.  ?  in 
30  mm.  ?  in  45  min.  ?  in  1  hr.  ?  12  hr.  ?  24  hr.  ?  in  1  wk.  ? 

198-200.  How  many  hours  in  1  wk.  ?  52  wks.  ?  365  days  ? 

201-207.  How  many  seconds,  Circular  Measure,  in  2'  ? 
in  8'  ?  in  15'  ?  in  2°  ?  in  8°  ?  in  15°  ?  in  1  cr.  ? 

208-211.  How  many  minutes  in  2°  ?  in  12°?  in  25°? 
in  1  cr,  ? 

n 


122  DENOMINATE  NUMBERS.  [CHAP.  XI. 

212-214.  How  many  units  in  6  doz.  ?  in  6  gross?  in  6 
great  gross  ? 

215-218.  How  many  units  in  5  score?  in  25  score?  in 
100  score  ? 

219-221.  How  many  pounds  in  5  bar.  flour?  in  25  ?  in 
100? 

222-225.  How  many  sheets  paper  in  5  quires  ?  in  1 
ream  ?  in  7  reams  ?  in  8  reams  ? 

226-233.  How  many  pence  in  100  fartli'ngs  ?  in  96/ar.  ? 
in  48/ar.  ?  in  112/ar.  ?  in  360/ar.  ?  in  24/ar.  ?  in  36/ar.  ? 
in  72/ar.  ? 

234-239.  How  many  shillings  in  144^.  ?  in  480<#.  ?  in 
96c?.  ?  in  1  40d.  -f  1  6/frr.  ?  in  lOOrf.  -f  80/ar.  ?  in  94d.  +  8/ar.  ? 

240-246.  How  many  pounds  sterling  in  360s,  ?  in  2405.  ? 
in  60s.?  in  4005.?  in  6405.?  in  585.  +  144rf.  ?  in  855.+ 


247-250.  How  many  sovereigns  in  1005.  ?  in  7505.  ?  in 
3725.  ?  in  2005.  ? 

251-262.  How  many  pennyweights  of  gold  in  96,07*.  ?  in 
300#r.?  in  lOO^r.  ?  in  10000T.  ?  in  1T280T.  ?  How  many 
ounces  of  gold  in  WQpwt.  ?  in  lOOOport.?  in  2  1  Opart.  ?  in 
400j?urt.  ?  How  many  pounds  of  gold  in  36502.  ?  in  72002.  ? 
in  17002.? 

263-265.  How  many  drams  in  213?  in  453  ?  in  1003  ? 

266-268.  How  many  ounces  in  483  ?  in  363  ?  in  6003? 

269-272.  How  many  pounds  in  1445  ?  in  3651  ?  in  100!  ? 
in  4605  ? 

273-276.  How  many  tons  in  400cw>£.  ?  in  lOOOcw^.  ?  in 
840cwtf.  ?  in  780cwtf.  ? 

277-281.  How  many  pounds  avoirdupois  in  160002.  ?  in 
36002.?  in500o2.?  in  36502.  ?  in  7llo2.  ? 

282-286.  How  many  yards  in  132/f.  ?  in  600//.  ?  in 
927/f.  ?  How  many  feet  in  100m.  ?  in  750m.  ? 


§  80.] 


UKDUCTIOX. 


123 


D< 

, 


287-289.  In  600  quarters  of  cloth,  how  many  yards  ? 
how  many  English  Ells  ?  how  many  French  Ells  ? 

290-292.  In  one  square  mile  of  land,  how  many  acres? 
ow  many  roods  ?  how  many  sq.  rods  ? 

293-295.  How  many  chains  in  500  links  ?  how  many 
feet  ?  In  660  feet  how  many  chains  ? 

296.  How  many  cords  of  wood  in  1280  cubic  feet  ? 

297.  In  a  pile  of  wood  4  feet  wide,  4  feet  high,  and  100 
feet  long,  how  many  cords  ? 

298.  How  many  bushels  in  WQpk.  ? 

299-300.  How  many  hours  in  SOOmin.  ?  in  1000mm.  ? 


REDUCTION  OF  DENOMINATE  QUANTITIES  * 

§  86.  When  the  quantity  is  to  be  reduced  from  a  higher 
to  a  lower  denomination,  the  process  is  called  Reduction 
Descending;  when  from  a  lower  to  a  higher,  Reduction 
Ascending. 


REDUCTION  DESCENDING. 


CASE  I. 

Reduce  £7  55.  IQd.  3far.  to  farthings. 

There  are  20s.  in  a  pound ;  therefore  7  X  20 
will  give  the  shillings  in  £7.  But  the  5s.  of 
the  given  quantity  are  also  to  be  reduced. 
Adding  these  to  the  shillings  already  found, 
•we  have  145s.  in  £7  5s.  Multiplying  145, 
the  number  of  shillings,  by  12,  because  there 
are  \2d.  in  a  shilling,  we  obtain  the  number 
of  pence  in  145s. ;  adding  the  IQd.,  we  have 
1750c?.  in  £7  5s.  IQd.  Multiplying  1750  by  4, 


£7  5s.  IQd.  Sfar. 
20 


140s. 

5s. 

145s. 

12 

1740(/. 
IQd 


7000/ar. 
ifar. 

7003/ar. 


*  The  terra  quantity  is  used  here  in  preference  to  the  term  number,  because  it 
may  include  denominate  fractions  as  well  {is  denominate  integers ;  and  because 
numbers  of  different  denominations  often  form  but  one  quantity. 


124  DENOMINATE  NUMBERS.  [CHAP.  XI. 

because  there  are  4  farthings  in  a  penny,  and  adding  the  3/ar.  to 
the  product,  we  have  7003/ar.  in  £7  5s.  10c?.  3/ar. 


CASE  II. 


Reduce  £^-Q  to  its  value  in  farthings,  or  in  the  fraction 
of  a  farthing. 

"We  proceed  as  in  Case  I.  Multiplying  £s^  by  20,  we  obtain  for 
a  product  the  value  of  £5^  in  the  fraction  of  a  shilling ;  that  is, 
j&n^=^j^*>=-jY*>  Multiplying  this  by  12,  we  obtain  its  equiva- 
lent value  in  the  fraction  of  a  penny  ;  that  is,  ^s.=±%d.=%d.  Mul- 
tiplying this  by  4,  we  obtain  its  equivalent  value  in  the  fraction  of  a 
farthing  ;  that  is,  $d.=%far.t  or  2§/cr. 

By  cancelation  the  process  becomes 


CASE    III. 

Reduce  -| yd.,  cloth  measure,  to  its  equivalent  value  in 
lower  denominations. 

"We  proceed  as  before,  multiplying  f ,  the  number  of  yards,  by  4, 
because  there  are  4  quarters  in  a  yard,  we  obtain  the  number  of 
quarters  equivalent  to  f  of  a  yard ;  that  is,  %yd.  =-l^-gr.=l^qr. 
Reducing  the  fractional  part  only  of  this  result,  we  have  ^qr.=$X4na. 
—±na.=2na.  Collecting  the  integers  we  have  Iqr.  2na.  for  the  an- 
swer. 

CASE  iv. 

Reduce  £0-778125  to  its  value  in  lower  denominations. 

We  first,  multiply  the  given  decimal  by  20,  as  in  cases  1  and  2, 
and  obtain  a  product  in  shillings,  and  the  decimal  of  a  shilling.  Re- 
ducing the  fractional  part  of  this  product  still  further,  we  obtain 


§  87.]         REDUCTION  OF  DENOMINATE  NUMBERS.  125 


its  equivalent  value  in  pence,  and  the  decimal 
of  a  penny.  Multiplying  the  decimal  part  of 
this  second  result  by  4,  to  reduce  it  to  farthings, 
we  obtain  3/ar.  for  a  final  product.  Collecting 
the  integers,  we  have  15s.  Qd.  3/ar.  for  the  an- 


15-5625005. 
12 

6-75000001. 
ewer. 


From  the  foregoing  examples,  we  may  deduce 


£0-778125 


20 


8-000000/ar. 


the  following  rule  for  the  reduction  of  higher  to  lower  denomi- 
nations. 

I.  Multiply  the  single  quantity  of  a  higher  denomination, 
or  the  highest  term  of  the  compound  quantity,  by  the  number 
of  the  next  lower  denomination  required  to  make  one  of  that 
higher  ;  the  product  will  be  in  the  lower  denomination. 

II.  To  this  product  add  the  term  (if  there  be  any)  in  the 
given  quantity,  which  is  of  the  same  denomination  as  the 
product,  and  multiply  as  before,  and  so  on. 

III.  The  final    product,  or   (if  the  single  quantity  be  a 
FRACTION)    the  integers,   if  any,  of  the  successive  products, 
taken  collectively,  will  give  the  result  required. 

§  87.     REDUCTION  ASCENDING. 

CASE  I. 

Reduce  7003  farthings,  to  pounds,  shillings,  pence,  and 
farthings. 

Obviously  the  process  of  Reduction  Ascending  is  the  reverse  of 
Reduction  Descending. 


In  7003/«r.,  there  can,  of  course,  be 
but  \  as  many  pence,  since  4  farthings 
make  1  penny.  Dividing,  then,  by  4, 
we  obtain  1750o*.  for  a  quotient,  and 
3/ar.  remainder.  Next  we  divide  the 


4 )  7003/ar. 
12  )  175007.       3/ar.  rem. 
210  )  1415s.     lOct  rem. 
£7  5s.  10(7.  3/ar. 


number  of  pence  by  12,  since  there  are 
12  pence  in  a  shilling,  and  obtain  145s.  for  a  quotient,  and  10e£.  re- 

11* 


126  DENOMINATE  NUMBERS.  [CHAP.  XI. 

mainder.  Lastly,  dividing  the  number  of  shillings  by  20,  we  obtain 
£7  for  a  quotient,  and  5s.  remainder.  We  have,  then,  for  a  total 
result,  £7  5s.  IQd  Sfar. 

CASE  II. 

Reduce  %far.  to  the  fraction  of  a  pound  sterling. 

We  proceed  as  in  Case  I.  Dividing  3  /err.  by  4,  we  obtain  for  a 
quotient  the  value  of  %far.  in  the  fraction  of  a  penny  ;  that  is, 
-Ad  Dividing  this  by  12,  we  obtain  for  a  quotient  the  value  of 
JjjG?.  in  the  fraction  of  a  shilling  ;  that  is,  -^^s.  Dividing  this  by  20, 
we  obtain  for  a  quotient  the  value  of  -,f  „«.  in  the  fraction  of  a  £  ; 

that  is,  2£^.=£o3oo==£iirW 

By  cancelation,  the  process  becomes, 


CASE  III. 

Reduce  2da.  Whr.  bmin.  to  the  fraction  of  a  week. 

As  there  are  60  minutes  in  1  hour,  to  reduce  any  number  of  ftin- 
utes  to  hours,  we  divide  by  60.  Then  5min.=s5^,  or  ^hr.  Adding 
to  this  quotient  the  107*  r.  of  the  quantity  to  be  reduced,  we  have 
IQj-hr.  Dividing  this  by  24,  to  reduce  it  to  days,  we  have  10TV*o. 
=  VV/<r.,  and  yj-:-24=i§J;  that  is,  10Ty*r.=lf  J<to.  Again,"  di- 
viding this  quotient  with  the  2da.  added  in,  by  7  to  reduce  it  to  weeks. 
^lda,=^lda.,  and  4^H-7=^TV,  that  is,  ^{da.=^^wk., 
the  required  fraction. 

CASE  IV. 

Reduce  15s.  6d.  3far.  to  the  decimal  of  a  pound  sterling. 

This  example  is  similar  to  the  preceding.  The  only  difference  is, 
that  the  answer  here  is  required  in  a  decimal  instead  of  a  common 
fraction. 

Dividing  3,  the  number  of  farthings,  by  4,  and  expressing  the  quo- 
tient decimally,  we  have  |=0-7o  that  is,  8far.=Q-tl5cL  Then  6ct 


§  87.]          REDUCTION  OF  DENOMINATE  NUMBERS,  127 

3/ar.=675<£  Dividing  by  12,  we  have  — ^-=0'5625  ;  that  is, 
6-75c£.=0-5625s.  Adding  to  this  quotient  the  15s.,  we  have  15s.  6d. 
3/ar.  =  15-56255.  Dividing  by  20,  we  find  -  — =0-778125  ; 

that  is,  15-5625s.=£0'778125,  for  the  required  decimal. 
The  work  may  be  conveniently  arranged  as  follows  : 
We  place  the  different  denominations  in     j         4|3/«r. 
a  column,  with  the  smallest  denomination  12'6'75d 

at  the  top ;  we  then  suppose  naughts  an-     j      2l0115>5625$. 


nexed  to  the  3  farthings,  and  divide  by  4, 


0-778125  of  a£. 


and  the  quotient,  which  must  be  a  decimal,  ' 
we  place  at  the  right  of  the  6d  ;  we  next  divide  6'75f/.  with  naughts 
annexed,  by  12,  and  the  quotient,  which  is  also  a  decimal,  we  pkce 
at  the  right  of  the  15s. ;  finally,  we  divide  the  15'5625s.  by  20.  In 
dividing  by  20,  we  cut  off  the  naught,  and  divide  by  2,  observing  to 
remove  the  decimal  point  one  place  to  the  left. 

From  the  foregoing  examples  we  may  deduce  the  following  rule 
for  the  reduction  of  lower  to  higher  denominations. 

I.  Divide  the  single  quantity  of  a  lower  denomination,  or 
the  lowest  term  of  the  compound  quantity,  by  the  number  ivhich 
is  required  of  such  denomination  to  make  one  of  the  next  high- 
er ;  the  quotient  will  be  in  that  higher  denomination. 

II.  To  this  quotient  add  the  term  (if  there  be  any)  in  the 
given  quantity,  which  is  of  the  same  denomination  as  the  quo- 
tient ;  and  divide  as  before,  and  so  on. 

III.  The  final  quotient,  or  (if  the  single  quantity  be  a 
whole  number)  the  final  quotient  with   the  intermediate  re- 
mainders, will  give  the  answer  required. 

PROMISCUOUS  EXERCISES  IN  REDUCTION  OF  DENOMINATE  QUAN- 
TITIES, INCLUDING  APPLICATIONS  OF  THE  TABLES. 

301.  In  £47  5s.  Id.  Ifar.  how  many  farthings  ? 

302.  In  118567  farthings,  how  many  pounds,  shillings, 
and  pence  ? 


128  DENOMINATE  NUMBERS.  [CHAP.  XI. 

303.  Reduce  £75  to  shillings. 

304.  Reduce  195.  Qd.  to  pence. 

305.  Reduce  25s.  3d.  2/ar.  to  farthings. 

306.  In  48926  grains,  Troy  Weight,  how  many  pounds, 
ounces,  pennyweights,  and  grains  ? 

307.  In  3605  pennyweights,  how  many  pounds,  ounces, 
and  pennyweights  ? 

308.  In  1000  ounces,  Troy  Weight,  how  many  pounds 
and  ounces  ? 

309.  In  4lb.  6oz.  ISpwt.  5gr.  how  many  grains? 

310.  In  100/6.  \gr.  how  many  grains? 

311.  In  4fo  53  13,  how  many  drams? 

312.  In  1000  grains,  Apothecaries'  Weight,  how  many 
ounces,  drams,  scruples,  and  grains  ? 

313.  In  11 5 21  grains,  Apothecaries'  Weight,  how  many 
pounds  ? 

314.  In  873450  drams,  Avoirdupois  Weight,  how  many 
tons? 

315.  Reduce  5cwt.  21/6.  4o2.  to  ounces. 

316.  Reduce  IT.  Icwt.  Idr.  to  drams. 

317.  Reduce  856702  drams  to  tons. 

318.  In  4355  inches,  how  many  yards  ? 

319.  In  248  miles,  how  many  inches  ? 

320.  How  many  inches  in  360  degrees,  of  69 £  miles  to 
each  degree,  which  is  the  circumference  of  the  earth,  nearly  ? 

321.  Reduce  12  Ells  French  to  nails. 

322.  Reduce  11  Ells  English,  3  quarters,  to  quarters. 

323.  Reduce  10  Ells  Flemish,  3  quarters,  1  nail,  to  nails. 

324.  Reduce  4  yards  to  quarters. 

325.  In  1000  nails,  how  many  yards  ? 

326.  How  many  inches  in  6  yards,  3  quarters? 

327.  How  many  square  inches  in  10  square  feet? 

328.  In  3  square  miles,  how  many  square  rods  or  poles? 


§  87.]         KEDXrCTION  OF  DENOMINATE  NUMBERS.  129 

329.  In  3  acres,  27  rods,  how  many  square  feet  ? 

330.  In  26025  square  feet,  how  many  square  roods? 

331.  In  70000  square  links,  how  many  square  chains  ? 

332.  Kfow  many  square  links  in  5  acres  ? 

333.  In  17  cords  of  wood,  how  many  cubic  feet  ? 

334.  In  17  tons  of  round  timber,  how  many  cubic  inches  ? 

335.  Reduce   17900345   cubic  inches  to  tons  of  hewn 
timber. 

336.  In  1000  cord  feet  of  wood,  how  many  cords  ? 

337.  In  19  cubic  feet,  how  many  cubic  inches  ? 

338.  In  16  hogsheads  of  wine,  how  many  gills  ? 

339.  In  10000  gills  of  wine,  how  many  barrels  ? 

340.  Reduce  2  pipes,  7  barrels,  3  quarts  of  wine,  to  pints. 

341.  Reduce  31752  gills  of  wine  to  barrels. 

342.  Reduce  201600  gills  to  tuns  of  wine. 

343.  Reduce  11  hogsheads  of  beer  to  pints. 

344.  In  100000  pints  of  beer,  how  many  hogsheads? 

345.  In  10  hogsheads,  1  quart,  1  pint  of  beer,  how  many 
pints  ? 

346.  In  36  bushels,  how  many  pints  ? 

347.  In  25  chaldrons,  29  bushels,  how  many  quarts  ? 

348.  In  10000  pints,  how  many  chaldrons? 

349.  In  1597  quarts,  how  many  bushels  ? 

350.  In  30  days,  how  many  seconds  ? 

351.  In  19  years,  of  365 J  days  each,  how  many  hours  ? 

352.  In  25  years  6  days,  how  many  seconds  ? 

353.  How  many  days  from  the  birth  of  Christ  to  Christ- 
mas,  1843,  allowing  the  years  to  consist  of  365  days  6 
hours  ? 

354.  A  person  was  born  May  3,  1795.     How  many  days 
old  was  he  May  3,  1821,  paying  particular  attention  to  the 
order  of  leap  year  ? 

055.  Suppose  a  person  was  born  February  29,  1796  ; 


130  DENOMINATE  NITMBEKS.  [CHAP.  XI. 

how  many  birthdays  will  he  have  seen  on  February  29,  1844, 
not  counting  the  day  on  which  he  was  born  ?* 

356.  In  3  signs  18  degrees,  how  many  seconds? 

357.  In  6  signs  9  degrees,  how  many  degrees  ? 

358.  In  1000'  how  many  degrees  ? 

359.  In  10000''  how  many  degrees  ? 

360.  Reduce  45°  45'  35"  to  seconds. 

361.  In  1000  things,  how  many  dozen  ? 

362.  How  many  buttons  in  6£  dozen  ? 

363.  In  80000  tacks,  how  many  gross? 

364.  In  three  score  and  ten  years,  how  many  years  ? 

365.  In  15  quires  of  paper,  how  many  sheets  ? 

366.  In  a  ream  of  paper,  how  many  sheets  ? 

367.  Reduce  J|£  of  a  yard  to  a  fraction  of  a  mile. 

368.  Reduce  |~J  of  a  gill  to  the  fraction  of  a  gallon. 

369.  Reduce  J-|§-  of  a  pound  to  the  fraction  of  a  ton. 

370.  Reduce  ^  of  a  mile  to  the  fraction  of  a  foot. 

371.  Reduce  J  of  J  of  f  of  a  yard  to  the  fraction  of  a  mile. 

372.  Reduce  ^  of  -J  of  J|  of  a  gallon  to  the  fraction  of 
a  gill. 

373.  Reduce  j-  of  |-  of  a  hogshead  of  wine  to  the  fraction 
of  a  gill. 

374.  Reduce  J  of  |  of  4J  yards  to  the  fraction  of  an 
inch.  • 

375.  Reduce  ^  of  -f^  of  a  farthing  to  the  fraction  of  a 
shilling. 

376.  Reduce  -/^  of  an  ounce  to  the  fraction  of  a  pound 
avoirdupois. 

377.  Reduce  -J  of  J  of  1  rod  to  the  fraction  of  an  inch, 
of  a  foot,  and  of  a  yard. 

*  It  must  be  recollected  that  the  year  1800  was  a  common  year,  having  no  SJ9th 
of  February. 


§  87.]         REDUCTION  OF  DENOMINATE  NUMBERS.  131 

378.  Reduce  J  of  f  of  1  hour  to  the  fraction  of  a  month 
of  30  days,  and  to  the  fraction  of  a  year  of  365  days. 

379.  Reduce  ^  of  1  yard  to  lower  denominations. 

380.  What  is  the  value  of  %  of  f  of  1  mile  ? 

381.  Reduce  f  of  -|  of  1  cwt.  to  lower  denominations. 

382.  What  is  the  value  of  -J  of  14  miles,  6  furlongs  ? 

383.  What  is  the  value  of  J-  of  f  of  2  days  of  24  hours 
each  ? 

384.  What  is  the  value  of  ^  of  |-  of  T3T  of  an  hour  ? 

385.  Reduce  if  ^  of  a  solar  day  to  lower  denominations. 

386.  What  is  the  value  of  }-|-g-  of  a  pound  avoirdupois? 

387.  What  is  the  value  of  T^  of  a  bushel  ? 

388.  What  is  the  value  of  -f^  of  a  year  of  365  days  ? 

389.  What  is  the  value  of  £  of  £  of  f  of  an  acre  ? 

390.  Reduce  £8  5s.  2d.  Iqr.  to  the  decimal  of  a  £. 

391.  Reduce  Sqr.  2na.  to  the  decimal  of  a  yard. 

392.  Reduce  1ft.  4m.  to  the  decimal  of  a  yard. 

393.  Reduce  3lb.  4t>2.  8pwt.  Igr.  Troy  to  the  decimal  of 
a  pound. 

394.  Reduce  QSfur.  Ird.  4yd.  2ft.  to  the  decimal  of  a  mile. 

395.  Reduce  3k.  30mm.  IQsec.  to  the  decimal  of  a  day. 

396.  Reduce  £3  55.  Od.  2far.  to  the  value  of  a  £. 

397.  Reduce  28  gallons  of  wine  to  the  decimal  of  a  hogs- 
head. 

398.  Reduce  4s.  6±d.  to  the  decimal  of  a  £. 

399.  Reduce  18s.  3%d.  to  the  decimal  of  a  £. 

400.  Reduce  3  pecks,  5  quarts,  and  1  pint  to  the  decimal 
of  a  bushel. 

401.  Reduce  llkr.  16m.  I5sec.  to  the  decimal  of  a  day. 

402.  Reduce  20  rods,  4  yards,  2  feet  and  6  inches  to  the 
decimal  of  a  furlong. 

403.  Reduce  42;?im.  36sec.  to  the  decimal  of  an  hour. 


132  DENOMINATE  NUMBERS.  [CHAP.  XI. 

404.  Reduce  30  days,  3  hours,  27  minutes,  30  seconds, 
to  the  decimal  of  a  year,  of  365*24224  days. 

405.  Reduce 5kr.  48min.  49*536sec.  to  the  decimal  of  a  day. 

406.  Reduce  0*9075  A  to  its  value  in  lower  denominations. 

407.  What  is  the  value  of  £0*125  ? 

408.  What  is  the  value  of  £0*66f  ? 

409.  Reduce  0*375  of  a  hogshead  of  wine  to  its  value  in 
lower  denominations. 

410.  Reduce  0-121212  of  a  year  of  365  days  to  its  value 
in  lower  denominate  numbers. 

411.  What  is  the  value  of  0*3355  of  a  pound  avoirdupois  ? 

412.  What  is  the  value  of  0'3322  of  a  ton  ? 

413.  What  is  the  value  of  0'2525  of  a  mile  ? 

414.  What  is  the  value  of  0*345  of  a  £  ? 

415.  What  is  the  value  of  0*121212  of  a  day  ? 

416.  What  is  the  value  of  0*3456  of  a  £  ? 

417.  What  is  the  value  of  0*9875  of  a  £  ? 

418.  What  is  the  value  of  0*24224  of  a  solar  day  ? 

419.  What  is  the  value  of  0*375  of  a  great  gross  ? 

420.  What  is  the  value  of  0*75  of  a  score  ? 

421.  What  is  the  value  of  0*485  of  a  quintal  of  fish  ? 
4-22.  What  is  the  value  of  0*3434  of  a  barrel  of  flour  ? 

423.  What  is  the  value  of  0*7575  of  a  barrel  of  pork  ? 

424.  What  is  the  value  of  0*985  of  a  quire  of  paper? 

425.  What  is  the  value  of  0'555  of  a  ream  of  paper  ? 

ADDITION  or  DENOMINATE  NUMBERS. 

§  88.  If  we  wish  to  find  the  sum  of  £6  5s.  3d.  Ifar., 
£7  Is.  Wd.  2far.,  £l  13s.  5d.,  £4  18s.  Qd.  2far.,  we  pro- 
ceed as  follows  : 

Placing  the  numbers  of  the  same  denomination  in  the  same  col- 
umn, we  add  the  column  of  farthings,  which  we  find  to  be  5.  But 


6531 
7  1  10  2 
1  13  5  0 

4  18     0     2 

£19  18s.  7d  Ifar. 


§  88.]  ADDITION  OF  DENOMINATE  NUMBERS.  133 

we  know  that  5  farthings  are  equivalent  to  1 
penny   and   1   farthing ;   we   therefore   write  s'      '  far' 

down  the  1  farthing  under  the  column  of  far- 
things, and  carry  the  penny  into  the  next  col- 
umn, whose  sum  thus  becomes  19  pence,  which 
is  the  same  as  1  shilling  and  7  pence  ;  we  write 
down  the  7  pence  under  the  column  of  pence, 
and  carry  the  shilling  to  the  column  of  kil- 
lings, whose  sum  then  becomes  38  shillings, 
which  is  the  same  as  1  pound  and  18  shillings ;  we  write  down  the 
18  shillings  under  the  column  of  shillings,  and  carry  the  pound  to 
the  column  of  pounds,  whose  sum  then  becomes  19  pounds ;  and  since 
pounds  is  the  highest  denomination,  we  write  down  the  whole. 
Hence  we  deduce  this  general 


RULE. 

I.  Place  the  numbers  so  that  those  of  the  same  denomina- 
tion may  stand  beneath  them  in  the  same  column. 

II.  Add  the  numbers  in  the  lowest  denomination ;  divide 
their  sum  by  the  number  expressing  how  many  of  such  de- 
nomination are  required  to  make    one   of    the  next  higher. 
Write  the  remainder  under  the  column  added,  and  carry  the 
quotient  to  the  next  column  ;  which  add  as  before. 

III.  Proceed  thus  through  all  the  denominations  to   the 
highest,  whose  sum  must  be  set  down  entire. 

EXAMPLES. 


(426.) 

£  s.   d. 

(427.) 

(428.) 

(429.) 

7  13  3 

£  s.  d. 

£   5.   d. 

Ib.  oz.pwt.gr. 

3  5  101 

11  0  5£ 

555 

10  10  10  10 

6  18  7 

2  4  4~ 

8  1  7f 

0  2  0  23 

0  2  5£ 

0  5  6| 

"  2  0  1£ 

3  0  17  0 

403 

1  3  4 

13  0  ll| 

2210 

17  15  41 

10  10  10 

666 

1  0  2  20 

12 


134 


DENOMINATE  NUMBEES.  [CHAP.  XI- 


(430.)                      (431.) 
Ib.  oz.pwt.gr.          Ib.  oz.pwt.  gr. 
6541           7305 
1   11  19  13         11     2  17  22 
0304         40     0     0  20 
8912           2  10  15  17 
4     4  19     0           0     6  18  16 

(432.) 
fij    §    3  3  gr. 
8  10  7  2  19 
10     0  6  0  10 
0     1  2  1   15 
5     1  2  1   15 
8     0  5  1    13 

(433.) 
ft  §   33 
2  11  6  0 
10     8  3  1 
14  10  2  2 
0     650 
7541 

(434.)                 (435.) 

(436.) 

(437.) 

3  3  gr.     ton.  cwt.  qr.  Ib,  oz.  dr.     cwt.  gr.  Ib.  oz.      L.  mi.  fur.  rd.  yd. 

1  0  18       10  18 

2  23  15     1 

4  3  20     5        1 

2     6  37  4 

2  1   15         1   15 

0     0  14  15 

5  0  12     3        6 

0     0  30  5 

3  2  13       12     0 

1     3     0  10 

1208        0 

1403 

400         0  13 

0  24     1  11 

0  3  24  13        2 

0110 

617         22 

2078 

1  2  20  10        3 

2     0  25  1 

(438.) 

(439.) 

(440.) 

(441.) 

rd.  yd.  ft.  in. 

yd.  qr.  na. 

E.  Fl.  qr  no. 

E.  E.  qr.  na. 

10     4     2     8 

15     1     2 

323 

422 

1305 

13     0     3 

15     1     2 

10     1     1 

8216 

20     2     2, 

920 

920 

1104 

030 

8     0     1 

13     0     2 

0219 

8     1     1 

10     0     0 

15     1     1 

(442.) 


(443.) 


(444.) 


(445.) 


Sq.yd.Sq.ft.Sq.in.  M.  A. 

R. 

P. 

8.  yd. 

s./t. 

S.in. 

a  s.  ft. 

100 

8 

130 

0 

100 

1 

30 

4 

26 

1000 

10 

120 

50 

0 

100 

10 

600 

2 

10 

1 

10 

1541 

8 

100 

10 

5 

0 

8 

40 

1 

12 

0 

20 

80 

2 

80 

0 

8 

143 

0 

0 

3 

2 

10 

17 

11 

0 

119 

13 

2 

8 

4 

4 

0 

20 

8 

25 

59 

12 

6 

§  89.]        SUBTRACTION  OF  DENOMINATE  NUMBERS.  135 


(446.) 

(447.) 

(448.) 

(449.) 

a 

3 

c.ft. 

7 

hhd.gal.  qt.pt. 
4  30  3  1 

tun. 
1 

pi.  hhd.  gal.  qt.  pt.  gi. 
1  1  37  3  1  3 

hhd.  gal. 
2  50 

gt.pt. 
3  1 

10 

4 

10 

25  0  1 

10 

0 

0 

50  0 

1  2 

10 

30 

1  0 

12 

1 

25 

020 

11 

0 

1 

13  1 

0  1 

11 

25 

0  1 

8 

6 

0 

60  0  1 

4 

1 

0 

25  2 

0  0 

25 

1 

1  0 

15 

3 

13 

45  3  0 

8 

0 

1 

18  0 

1  3 

6 

52 

3  1 

(450.) 

(451.) 

(452.) 

(453.) 

bar.  gal.  qt. 
10  30     1 

ch.  bu.  pk.  qt.  pt. 
1  30     3     7     1 

bu.  pk.  qt.  pt. 
10     1      1      1 

da.  hr.  m.  sec, 
15  18  50  49 

6  20     0 

0  30     2     3     0 

2360 

1   13  59  59 

1     5     2 

10  19     1     0     1 

5230 

4  23     0     2 

10     0     3 

5  10     2     4     0 

8001 

10  11     1     4 

4  25     1 

44051 

15    '2     4     0 

0     2  10  15 

(454.)  (455.)  (456.)  (457.) 

vk.  da.  hr.  m.  sec.        cr.    8.    °      '     "          s.    °       '          °      ' 


I 

2  13  40  30 

1 

8 

25 

40 

35 

1 

25 

2 

13 

10  19 

2 

6  10  8  3 

0 

11 

1 

2 

43 

0 

18 

50 

1 

40  35 

0 

5  22  55  45 

1 

0 

29 

59 

0 

2 

5 

39 

2 

48  39 

2 

3  4  1  15 

0 

1 

10 

13 

5 

0 

4 

4 

0 

30  40 

1 

2450 

0 

2 

5 

4 

3 

4 

15 

10 

10 

45  45 

SUBTRACTION  OF  DENOMINATE  NUMBERS. 
89.  Subtract  £15  135.  lOdL  from  £20  5s.  Sd. 


We  place  the  numbers  of  the  subtrahend  di- 
rectly under  the  numbers  of  the  same  denomina- 
tion in  the  minuend.  We  cannot  subtract  10c?. 
from  Sd. ;  we  therefore  increase  the  Sd.  by  \1d. 
making  20d. ;  then  subtracting  10c?.  from  the  20d. 
we  have  the  difference  10c?.,  which  we  write  under 
the  column  of  pence.  Having  added  12</.  to  the  minuend,  we  must 
equally  increase  the  subtrahend,  which  we  do  by  adding  1«.  (the 


£  s.      d. 

20  58 

15  13     10 

4  11     10 


136  DENOMINATE  NUMBERS.  [CHAP.  XI, 

same  as  the  12d)  to  the  13s.,  making  14s.  This  cannot  be  subtract- 
ed from  5s.;  we  therefore  increase  the  5s.  by  20s.,  making  25s. 
Now,  subtracting  14s.  from  25s.  we  have  11s.,  which  we  write  under 
the  column  of  shillings.  Before  subtracting  the  pounds,  we  add  £1 
to  £15  to  compensate  for  the  20s.  added  to  the  5s.,  and  then  say 
£16  from  £20  leaves  £4. 
Hence,  we  have  this  general 

RULE. 

I.  Place  the  less  number  under  the  greater,  so   that  the 
same  denominations  may  stand  in  the  same  column. 

II.  Begin  at  the  right,  and  subtract  each  number  in  tlie 
lower  line  from  the  one  directly  above  it,  and  set  the  remain- 
der below. 

III.  If  any  number  in  the  lower  line  is  greater  than  the 
one  above  it,  add  so  many  to  the  upper  number  as  make  one 
of  the  next   higher  denomination ;  then   subtract  the  lower 
number  from  the  upper  one  thus  increased,  and  set  down  the 
remainder.       Carry  1,  expressing  the  increase  of  the  upper 
line,  to  the  next  number  in  the  lower  line  ;  after  which  sub- 
tract this  number  from  the  one  above  it,  as  before  ;  and  thus 
proceed  till  all  the  numbers  are  subtracted. 

PROOF. 

If  the  work  be  right,  the  difference  added  to  the  subtra- 
hend will  equal  the  minuend,  as  in  simple  subtraction. 

EXAMPLES. 

(458.)  (459.)  (460.) 

T.  cwt.  qr.  Ib.  oz.  dr.  A.  R.  P.  EJ  %  3  3  gr. 

13  18  1  20  0  13  69  3  25  24  7  2  1  16 

10   0  3  21  12  0  10  0  38  16  10  3  2  17 


§  89.]       SUBTRACTION  OF  DENOMINATE  NUMBERS.  137 


(461.) 
L.  mi.  fur.  rd. 
16  2  7  39 
5078 

(464.) 

(462. 
E.  Fr.  qr. 
10   5 
5   1 

)           (463.) 
na.      ch.  bu.  pic.  qt.  pt. 
0      30  10  1  1  0 
3      10   8  3  6  1 

V465.) 

(466.) 

tun.  pi.  hhd.  gal. 
10  1   1  50 

qt.      da. 
1      100 

hr.   m.  sec. 
10   0   1 

yr.  mo.  wk.  da. 
17  8  3  1 

(467 

0 

0 

60 

3       60 

0 

40  45 

4  1 

2  6 

(468.) 

(469.) 

(470.) 

m.  fur. 
60  0 

rd. 
0 

C.  S.ft. 
45  126 

G.  Cord  ft. 
100   6 

£ 
50 

0 

d. 
\ 

40  7 

39 

10  127 

80   7 

30 

10 

10 

(471.) 

(472 

.)            (473.) 

yr. 
1838 

mo. 
9 

da. 
6 

yr.   mo. 
1839  2 

da. 
4 

yr. 
1840 

mo. 
9 

da. 
15 

1837 

10 

1 

1838 

9 

6 

1837 

10 

1 

(474.) 

(475.) 

(476.) 

yr. 
1840 

mo. 
3 

da. 
5 

yr.   mo. 
1850  9 

da. 
3 

yr. 
1850 

mo. 
11 

da. 
2 

1836 

4 

1 

1796 

6 

7 

1776 

6 

4 

EXERCISES  IN  ADDITION    AND  SUBTRACTION. 

477.  Bought  20  yards  of  broadcloth  for  £18  5s.  3d.,  30 
pounds  of  feathers  for  £8  25.  4d.,  100  yards  carpeting  for 
£45  17s.  8d.,  10  pieces  of  cotton  cloth  for  £8  18*.  Id.,  50 
yards  of  calico  for  £2  05.  Wd.     What  was  the  cost  of  the 
whole  ? 

478.  Bought  four  hogsheads  of  sugar,  weighing  as  fol- 
lows:    1st  weighed  8cwt.  Iqr.  23lb.  1002.  ;     2d  weighed 

12* 


138  DENOMINATE  NUMBERS.  [CHAP.  XI. 

Qcwt.  2qr.  Gib.  3oz. ;  3d  weighed  Wcivt.  Oqr.  Olb.  8oz. ;  4th 
weighed  8cwt.  3qr.  23lb.     How  much  did  the  four  weigh  ? 

479.  A  man  owns  three  farms  :  the  first  contains  69  acres, 
3  roods,  10  rods;  the  second  contains  100  acres,  5  rods; 
the  third  contains  150  acres,  2  roods.     How  many  acres  are 
there  in  all  ? 

480.  Suppose  a  note  given  August  3d,  1838,  to  be  paid 
November  10th,  1843.     How  long  was  the  note  on  interest, 
if  we  count  30  days  to  the  month  ?  and  how  long  if  the  time 
is  accurately  computed  ? 

481.  A  person  buys  I5cwt.  3qr.  20£6.  of  sugar,  and  sells 
Wcwt.  Oqr.  11/6.     How  much  remains  unsold  ? 

482.  From  a  piece  of  cloth  containing  3*lyd.  3qr.  2nd. 
there  has  been  taken  at  one  time  6yd.  Iqr.,  at  another  time 
IQyd.  3qr.  3na.     How  much  remains  ? 

483.  From  a  pile  of  wood  containing  100  cords,  I  sold  at 
one  time  10(7.  1005.  ft. ;  at  another  time  18(7.  59$.  ft. 
How  many  cords  remain  unsold  ? 

484.  A  farmer  raises  WQbu.  3pk.  2qt.  of  wheat  from  one 
field  ;  8lbu.  \pk.  Iqt.  \pt.  from  another  field  ;  he  sells  53bu. 
to  one  person,  and  37 bu.  2pk.  Iqt.  to  another  person.     How 
many  bushels  has  he  remaining  ? 

485.  Bought  5  loads  *  of  coal.     The  first  weighed  2056 
pounds ;  the  second,  2250  ;  the  third,  2240  ;    the  fourth, 
2310  ;  the  fifth,  2330.     What  was  the  entire  weight  ?    And 
how  many  tons  of  2000  pounds  each  ? 

486.  A  person  engages  to  build  100  rods  and  10  feet  of 
stone  fence  :  at  one  time  he  builds  1 7  rods,  5  feet ;  at  an- 
other time  37  rods,  15  feet.     How  much  still  remains  to  be 
built? 

487.  How  much  cloth  in  3  pieces,  measuring  as  follows : 
first  piece  37  yards,  3  quarters,  1  nail ;  second  piece  41  yards, 
1 J  Flemish  Ells  ;  third  piece  43  yards,  l|  English  Ells  ? 


§  90.]      MULTIPLICATION  OF  DENOMINATE  NUMBERS.      139 

488.  Bought  3  loads  of  wood  :  the  first  was  8  feet  long, 
4  feet  wide,  and  3  feet  high  ;  the  second  was  7  feet  long, 
4  feet  wide,  and  2  feet  high ;  the  third  was  9  feet  long,  3 
feet  wide,  and  3  feet  high.  How  many  solid  feet  in  the 
whole  ?  How  many  cord  feet,  and  how  many  cords  ? 

"489-491.  George  Washington  was  born  Feb.  22,  1732  ; 
John  Adams  Oct.  19,  1735.  How  much  earlier  was  the 
birth  of  the  former  than  that  of  the  latter  ?  How  long  a 
time  from  the  birth  of  each  to  January  1,  1851  ? 

492.  William  Shakspeare  was  born  April  23,  1564.    How 
long  since  was  that,  estimating  from  January  1,  1851  ? 

493.  How  long  from  the  birth  of  Milton,  Dec.  9,  1608, 
to  the  birth  of  George  Washington  ? 

494.  The  latitude  of  New  Orleans  is  29°  57'  30",  that 
of  Portland  is  43°  36 '.     What  is  the  difference  in  latitude 
of  these  two  places  ? 

495-496.  The  latitude  of  New  York  is  40°  42'  40".  How 
far  is  it  north  of  New  Orleans  ?  and  how  far  south  of  Port- 
land ? 

497-499.  The  latitude  of  Boston  is  42°  21'  23".  How 
far  is  it  north  of  New  York  ?  how  far  north  of  New  Or- 
leans ?  how  far  south  of  Portland  ? 

500.  Henry  Jenkins,  of  England,  died  Dec.  8,  1670,  at  the 
advanced  age  of  168  years,  9  months.    When  was  he  born  ? 

501.  America  was  discovered  by  Columbus  Oct.  14,  1492. 
How  long  was  that  before  the  landing  of  the  New  England 
Pilgrims,  Dec.  20,  1620  ? 

MULTIPLICATION  OF  DENOMINATE  NUMBERS. 

§  90.  Multiply  £13  5s.  IQd.  by  5. 

First,  -we  say  5  times  10c?.  is  50d.,  which  equals  4s.  and  2c£. ;  we 
set  down  the  2d.  and  reserve  the  4*.  We  then  say  5  times  5s.  equals 


140  DENOMINATE  NUMBERS.  [CHAP.  XI. 

25s.,  to  which,  adding  the  4s.  we  have  29s.,  which  £     s.      d. 

equals  £1  9s. ;  we  set  down  the  9s.  and  reserve  13     5     10 

the  £1.     Finally,  we  say  5  times  £13  is  £65,  to  5 

which  adding  the  £1,  we  have  £66  ;  this  being  ~~     ~ 

the  highest  denomination,  we  set  it  down  entire. 
Hence  this  general 

,  RULE. 

Commence  at  the  right  hand,  and  multiply  the  number  in 
each  denomination  by  the  multiplier.  Divide  each  product 
by  the  number  expressing  how  many  of  the  denomination  of 
suck  product  are  required  to  make  one  of  the  next  higher  de- 
nomination. Write  the  remainder,  if  any,  under  the  num- 
ber multiplied,  and  carry  the  quotient  to  the  next  column. 

The  entire  product  of  the  highest  denomination  must  be 
set  down. 

EXAMPLES. 

502-507.  Multiply  £10  105.  IQd.  by  3  ;  by  5  ;  by  6  ;  by 
7 ;  by  8  ;  by  9. 

508-514.  Multiply  8cwt.  Oqr.  2lb.  4oz.  5dr.  by  3  ;  4  ;  5  ; 
6  ;  7  ;  8  ;  9. 

515-523.  Multiply  8cwt.  Gib.  Idr.  by  3  ;  4  ;  5 ;  6  ;  7  ; 
8;  9;  11;  12. 

524-529.  Multiply  Sgal.  3qt.  Ipt.  3gi.  by  35  ;  42  ;  30  ; 
54;  81;  49. 

NOTE. — The  multipliers  in  the  preceding  example  may  be  regard- 
ed as  composite  numbers. 

530.  In  3  hogsheads  of  sugar,  each  containing  Wcwt.  3qr. 
5lb.,  how  many  hundred  weight? 

531.  How  much  cloth  will  it  take  for  7  suits  of  clothes, 
if  each  suit  require  1yd.  3qr.  Ina.  ? 

532.  How  much  wood  can  a  horse  draw  in  13  loads,  if  he 
draw  1  C.  IQS.ft.  at  each  load  ? 


§  90.]     MULTIPLICATION  OF  DENOMINATE  NUMBERS.       141 

533.  How  long  will  it  take  a  man  to  saw  6  cords  of  wood, 
if  he  employ  Ihr.  3Qmin.  45sec.  to  saw  one  cord,  allowing 
10  working  hours  for  each  day  ? 

534.  The  circumference  of  a  wheel  is  15  feet  2  inches. 
What  distance  will  this  wheel  measure  on  the  ground,  if  it 
is  rolled  over  365  times  ? 

535.  Allowing  the  year  to  coilist  accurately  of  365  days, 
5  hours,  48  minutes,  49^  seconds,  what  will  be  the  true 
length  of  1843  years? 

536.  What  will  35cwt.  of  cheese  cost,  at  155.  Qd.  per  hun- 
dred weight  ? 

537.  How  much  brandy  in  84p.,  each  containing  128gal. 
2qt.  Ipt.  3gi.  ? 

538.  In  21  loads  of  wood,  each  1C.  Ic.ft.,  how  many 
cords  ? 

539.  Suppose  the  piston-rod  of  a  steam-engine  to  move 
3ft.  4jm.  at  each  stroke.     Through  what  distance  will  it 
move  in  making  1000  strokes  ? 

540.  Bought  as  follows  : 

Ib.  s.     d. 

18  of  green  tea,  at         12     3  per  pound. 

12  of  raisins,  "  12"" 

27  of  loaf  sugar,  "  14"      " 

15  of  English  currants,  "  23"      " 

14  of  citron,  "  36"      " 
What  is  the  amount  of  the  whole  purchase  ? 

541.  What  is  the  amount  of  the  following  bill  of  goods? 

£     s.    d. 

15  yards  of  broadcloth,  at    1     3     6  per  yard. 
12      "      "  silk,  18     3     "      " 
20       "      "  calico,           "  19"      " 
24      "      "  sheeting,      "  13"      " 
22      "      "  muslin,         "  34"      " 


142 


DENOMINATE  NUMBERS. 


[CHAP, 


DIVISION  OF  DENOMINATE  NUMBERS. 


17  )  £100  10s.  3d:  (  £5 
_86 

~15 

20 

17  )  310  (18s. 
17 
140 
136_ 
4 
12 

17  )  51  (  8d. 
51 

Collecting,  we  have 
£5  ISs.  3d 


§  91.  Divide  £100  10s.  3d.  equally  among  17  men. 

First,  we  say  17  in  £100,  is  contained 
5  times  and  £15  remaining;  and  since 
these  £15,  as  well  as  the  10s.,  are  yet  to 
be  divided  among  the  17  menjpre  reduce 
the  pounds  to  shillings,  and  add  the  10s., 
making  310s. ;  we  find  17  to  be  contained 
18  times  in  310s.  with  4s.  remainder.  "We 
reduce  the  4s.  to  pence,  and  add  the  3d, 
making  5 Id,  which  divided  among  the 
17  men,  gives  3d  each. 

NOTE. — "We  do  not  divide  100  pounds 
by  17  men,  which  is  impossible  ;  but  we 
separate  £100  into  17  equal  parts.  Each 
part  is  expressed  by  the  quotient,  and 
contains  £5,  §  71. 

Or,  adhering  to  the  general  definition  of  Division,  we  suppose  a 
pound  set  apart  for  each  man,  and  then  find  how  many  times  £17, 
the  number  thus  set  apart,  is  contained  in  £100 ;  the  quotient  will 
be  an  abstract  number.  The  answer  will,  of  course,  be  as  many 
pounds  to  each  man  as  there  are  parcels  of  £17  in  £100  ;  that  is,  as 
there  are  units  in  the  quotient. 

Had  the  divisor  been  a  single  digit,  the  work  might  have  been 
performed  by  Short  Division. 

We  therefore  have  this  general 

RULE. 

Place  the  divisor  on  the  left  of  the  dividend,  as  in  Simple 
Division.  Divide  first  the  number  of  the  highest  denomina- 
tion by  the  divisor.  Reduce  the  remainder,  if  any,  to  the 
next  lower  denomination  (adding  the  number  in  the  dividend 
of  that  denomination),  and  divide  the  sum  by  the  divisor, 
and  so  on. 

If  there  is  a  final  remainder,  express  its  division  in  a 
fractional  form,  and  annex  it  to  the  quotient. 


§  91.]  DENOMINATE  NUMBERS.  143 


EXAMPLES. 

542-548.  Divide  25yd.  3qr.  Ina.  by  3  ;  4  ;  5  ;  6  ;  7 ; 
8;  9. 

549-562.  Divide  2lcwt.  3qr.  20/6.  13oz.  9dr.  by  2  ;  3  ; 
4;  5;  6;  7;  8;  9;  by  27  ;  63 j  81  ;  64;  42  ;  49. 

NOTE. — The  pupil  will  observe  that  these  latter  divisors  are  com- 
posite numbers. 

563-567.  Divide  Wlb.  Soz.  IQpwt.  3gr.  by  13  ;  19  ;  23  ; 
29;  31. 

568.  A  man  has  23  days  allowed  to  walk  lOQmi.  4fur. 
SOrd.  \\yd.  2ft.  How  far  must  he  walk  each  day  ? 

569-572.  Divide  10  tuns  2hhd.  l^lgal.  2pt.  by  67  ;  51-4. 
Ift.  IIP.  by  51  ;  4gal.  2qt.  by  144  ;  £113  13*.  4d.  by  31. 

573.  Divide  673141c?a.  9kr.  58m.  24sec.  by  1843. 

574.  Divide  Imi.  255ft.  Win.  by  365. 

575.  Bought  15  sheep  for  £5  12s.  6d.     How  much  did 
one  sheep  cost  ? 

576.  If  24yds.  of  cloth  cost  £18  6s.,  how  much  is  that 
per  yard  ? 

577.  From  a  piece  of  cloth  containing  128yc?s.  Iqr.,  a 
tailor  made  18  coats,  which  took  one-third  of  the  whole 
piece.     How  many  yards  did  each  coat  contain  ? 


QUESTIONS  INVOLVING  THE  FOUR  PRECEDING  RULES. 

578.  Twenty-four  men  agree  to  construct  7mi.  Ifur.  24rd. 
of  road  :  after  completing  |-  of  it,  they  employ  8  more  men. 
What  distance  does  each  man  construct  before  and  after  the 
8  men  were  employed  ? 

579.  A  silversmith  has  seven  tea-pots,  each  weighing  lib. 
Soz.  I3pwt.  llgr.     What  is  the  whole  weight  ? 


144  DENOMINATE  NUMBERS.  [CHAP.  XL 

580.  A  farmer  has  1000  bushels  of  apples,  which  he  puts 
into  350  barrels.     How  many  does  each  barrel  hold  ? 

581.  If  it  require  1  sheet  of  paper  to  print  24  pages  of 
a  book,  how  many  reams,  allowing  1 8  quires  to  the  ream, 
will  it  take  to  print  3000  copies,  of  250  pages  each  ? 

582.  An  estate  worth  £2570  is  to  be  divided  as  follows: 
the  widow  has  one-third  of  the  whole,  the  remainder  is  to 
be  divided  equally  between  seven  children.     How  much  does 
the  widow  receive,  and  how  much  does  each  child  have  ? 

583.  Divide  100  acres,  3  roods,  8  rods  of  land,  between 
four  persons,  A.,  B.,  C.,  and  D.,  so  that  A.  shall  have  one- 
sixth  of  the  whole,  B.  one-fourth  of  the  remainder,  C.  one- 
third  of  what  then  remains,  and  D.  the  rest.     How  much 
does  each  one  have  ? 

584.  A.,  B.,  C.  and  D.,  having  I3cwt.  Iqr.  4lb.  of  sugar, 
they  agree  to  divide  it  as  follows  :  A.  is  to  have  one-half 
of  the  whole,  B.  is  to  have  one- third  of  the  remainder,  C.  is 
to  have  one-fourth  of  what  then  remains,  and  D.  is  to  take 
what  is  left.     What  were  their  respective  portions  ? 

585.  What  is   the  weight  of   the  following  coins :    10 
guineas,  each  weighing    5  pwt.  9 J  grains ;    7  sovereigns, 
each  weighing  1  pwt.  8  J  grains  ? 

586.  What  is  the  weight  of  13  crowns,  each  weighing 
18  pwt.  4T4T  grains;    14  shillings,   each  weighing  3 pwt. 

grm  •  9  sixpences,  each  weighing  1  pwt.  19T7T  gr.  ? 

587.  In  one  eagle  there  are  232-j^  grains  of  pure  gold, 
grains  of  silver,  and  12T9Q  grains  of  copper,  and  the 

same  proportions  of  gold,  silver,  and  copper  for  all  other 
American  gold  coin.  In  10  eagles,  7  half-eagles,  5  quarter- 
eagles,  how  many  grains  of  gold,  silver,  and  copper  ? 

588.  One  pound  of  pure  gold  is  sufficient  for  how  many 
dollars  of  gold  coin,  if   it  require  23 -22  grains  for  one 
dollar? 


§  91.]  DENOMINATE  NUMBERS.  14:5 

589.  One  pound  of  pure  silver  is  sufficient  for  how  many 
dollars  of  silver  coin,  if   it  require  371 '25  grains  for  one 
dollar  ? 

590.  The    aggregate  ages  of    35  individuals  amount  to 
964yr.  llmo.  3wk.  Qda.  13Ar.  47mm.  35sec.     What  would 
be  the  age  of  each  of  the  35,  supposing  their  ages  were  all 
equal  ? 

591.  For  13  successive  years  a  man  peddled  his  wares 
from  Nov.  7  to  March  16  inclusive,  in  the  Southern  States ; 
from  March  21  to  June  29,  in  the  Western  States ;  from 
July  14  to  Oct.  2,  in  the  Northern  States ;  and  20  days  of 
each  year  he  passed  in  the  Middle  States.     How  long  was 
he  in  each  section  of  the  country  ?  and  how  long  in  all  ? 

592.  If  one  solar  year  is  365da.  5kr.  48mm.  47'588sec., 
what  will  be  the  length  of  1000  years  ? 

593.  The  moon  occupies  about  2 9 '5 3  days,  on  an  average, 
from  change  to  change.     How  long  will  be  required  to  make 
235  changes  or  lunations  ? 

594.  The  moon  moves  through  an  entire  circumference 
of  360°  in  27c?a.  7Ar.  43mm.  1  l'5sec.     How  far  does  it  move 
each  day  ? 

595.  If  £1000  10s.  5d.  Sfar.  be  divided  equally  among 
23  individuals,  how  much  will  each  receive  ? 

596.  Bought  5  casks    of    sugar,  each  weighing  400/6. 
13oz.  ;  4  casks  of  375/6.  lOoz.  each.     What  was  the  weight 
of  the  9  casks  ? 

597.  If  a  ship  sail  3000  miles  in  lOda.  13hr.,  what  is  the 
average  hourly  rate? 

598.  If  we  estimate  one  degree  of  the  earth's  equator 
at  69£  miles,  what  will  be  the  length  of  an  arc  of  14°  18'  ? 

599.  If  the  circumference  of  the  earth  at  the  equator  is 
24899  miles,  through  what  distance  is  a  point  of  the  equator 
carried  each  hour  by  the  earth's  daily  revolution  ? 

13 


146  DENOMINATE  NUMBERS.  [CHAP.  XL 

600.  If  each  individual  of  a  city  of  50000  inhabitants 
requires  4'75  gallons  of  water  daily,  how  much  will  be  an- 
nually required  to  supply  the  city  ? 


DUODECIMALS. 

§  92.  DUODECIMALS*  are  a  kind  of  denominate  numbers, 
whose  denominations  increase  or  decrease  in  a  twelve-fold 
ratio.  They  are  applied  to  the  measurement  of  surfaces  and 
solids. 

The  denominations  of  Duodecimals  are  the  foot  (/.),  which 
is  the  unit  ;  the  inch  or  prime  ('),  which  is  T*^  of  the/.  ;  the 
second  ("),  which  is  j1^  of  the  prime  ;  the  third  ('"),  which 
is  yJj  of  the  second,  &c.  The  accents  which  are  used  to 
distinguish  the  denominations  below  feet  are  called  indices. 

The  Addition  and  Subtraction  of  duodecimals  are  per- 
formed like  addition  and  subtraction  of  other  denominate 
numbers.  It  need  only  be  remembered  that  12  of  any  de- 
nomination make  1  of  the  next  higher. 

EXAMPLES. 

(601.)  (602.) 

17/.     7'     8"  365/.     I7     7"  9'" 

25/     0',    2"  521/.   10'  10"  11"' 

30/.   10'  11"  605/     8'     8"  1 

29/.     6'     6"  731/.     3'     0"  8 


'" 


603.  What  is  the  sum  of  3/.  6'  4",  8/.  3'  4",  9/.  1'  3", 
and  10/.  10' 10"? 

604.  What  is  the  sum  of  100/.  8'  8",  135/.  0'  1",  65/ 
9'  2",  45/.  3'  3",  and  200/.  6'  6"  ? 

*  From  a  Latin  word,  duodecim,  meaning  tirefae. 


§  93.J  DUODECIMALS.  147 

(605.)  (606.) 

From        8V/.     3'      4"  100/.   10'  10" 

Subtract   35/     8'     9"  90/.     6'     3" 

607.  From  25/  6'  6"  subtract  18/.  9'  10". 

608.  From  100/.  subtract  58/.  2'  I". 

609-612.  WJiat  is  the  sum,  and  what  is  the  difference  of 
37/  11' 3"  and  13/.  I'll"?  of  99/  9'  9"  and  3 1/.  10'  10"  ? 

613-615.  From  100/.  subtract  11/11'  1";  from  the  re- 
mainder subtract  ll/  11'  1" ;  and  from  this  remainder  sub- 
tract ll/  11'  1".  What  are  the  three  successive  remain- 
ders? 

616-619.  What  is  the  sum  and  difference  of  47/  1'  1"  1'" 
and  13/  11'  11"  11'"?  of  10 1/  10'  10"  10'"  and  19/  3' 
3"  1'"? 

620.  What  is  the  sum  of  6/  3'  4",  4/  4'  0"  4'",  and 
13/  6'  6"  2/;/  ? 

MULTIPLICATION  OF  DUODECIMALS. 

§  93.  Suppose  we  wish  to  multiply  14/  7'  by  2/  3',  we 
should  proceed  as  follows  : 

14/     7' 
2/     3' 

3/     7'     9" 
29/     2' 


.   32/     9'     9" 

We  begin  on  the  right  hand,  and  multiply  the  multiplicand  througli, 
first  by  the  primes  of  the  multiplier,  then  by  the  feet  of  the  multi- 
plier ;  thus,  3'X7'=f32XT72=r1fV  of  a  foot>  which  is  21"=!' 9";  we 
write  down  the  9",  and  reserve  the  1'  for  the  next  product ;  again, 
14/.X3'=14X  -,32-=Tl  of  a  foot»  which  is  42';  now  adding  in  the  1', 


148  DENOMINATE  NUMBERS.  [CHAP.  XI. 

which  was  reserved  from  the  last  product,  we  have  43'=3/.  7',  which 
we  write  down,  thus  finishing  the  first  line  of  products. 

Again,  we  have  2/.X7'=2XT72=ff  of  a  foot,  which  is  14'=1/.  2'  ; 
we  write  the  2'  under  the  primes  of  the  line  above,  and  reserve  the 
If.  for  the  next  product  ;  2/.Xl4/.=28/.,  to  which,  adding  in  the  I/. 
reserved  from  the  last  product,  we  have  29/.,  which  we  place  under- 
neath the  feet  of  the  line  above.  Taking  the  sum,  we  find  32/.  9'  9" 
for  the  answer,  or  32/.+T°2/.-{-  fff/. 

It  will  thus  be  seen  that,  in  the  multiplication  of  Duode- 
cimals, the  sum  of  the  indices  of  the  factors  is  the  number 
of  the  indices  of  the  product,  just  as  in  decimals  the  sum 
of  the  decimal  places  in  multiplier  and  multiplicand  forms 
the  number  of  decimal  places  in  the  product.  Or  each  in- 
dex (')  might  be  regarded  as  denoting  the  factor  12  in  the 
denominator  of  a  fraction,  of  which  the  number  having  the 

2 
index  is  the  numerator.     Thus,  lf=^f.  ;  2"=—  —  —f-  = 


A/-  =mf-  =  i  "  ;  2"  x 


12302"    i2 

"2T81T  T2  /-  —  I  O'""-     The  foot,  being  the  unit,  or  integer,  has 
no  index. 

NOTE.  —  It  must  be  remembered  that  in  Duodecimals,  when  not 
used  to  express  linear  measure,  but  surfaces  or  solids,  the  foot  con- 
tains 144,s<7.  in.  or  1728s.  in.  Consequently,  in  the  measurement  of 
surfaces,  2'  would  be  equal,  not  to  2sq.  in.,  but  to  ^  of  1445^.  in.,  or 
24:sq.  in.  In  the  measurement  of  solids,  2'  would  be  T2^  of  1728s.  in.= 
288s.  in.  The  prime  (square  measure)  is  a  strip  of  surface  of  1  inch 
wide  and  12  inches  long  ;  the  prime  (solid  measure)  is  a  slab  1  inch 
thick,  12  long,  and  12  broad. 

From  what  has  been  said,  we  infer  the  following 

RULE. 
Place  the  several  terms  of  the  multiplier  under  the  corre- 


§  93.]  DUODECIMALS.  149 

sponding  ones  of  the  multiplicand.  Beginning  at  the  right 
hand,  multiply  the  several  terms  of  the  multiplicand  by  the 
several  terms  of  the  multiplier  successively,  placing  the  right- 
hand  term  of  each  of  the  partial  products  under  its  multiplier. 
To  each  product- term  annex  as  many  indices  as  are  in  both 
its  factors.  The  sum  of  the  partial  products  will  be  the  re- 
sult required. 

EXAMPLES. 

621.  What  is  the  product  of  3/.  7'  2"  by  7/  6'  3"? 
622-623.  Multiply  7/.  8'  by  6/.  4'  3" ;    6/.  9'  1"  by 
4f.  2'. 

624.  What  is  the  area  of  a  marble  slab  whose  length  is 
If.  3',  and  breadth  2/.  11'  ? 

625.  How  many  square  feet  are  contained  in  the  floor  of 
a  hall  37/.  3'  long,  by  10/.  7'  wide  ? 

626.  How  many  square  feet  are  contained  in  a  garden 
100/.  6'  in  length,  by  39/.  7'  in  width  ? 

627.  How  many  yards  of  carpeting,  one  yard  in  width, 
will  it  require  to  cover  a  room  16/.  5'  by  13/.  7'  ? 

628.  How  many  yards  of  Brussels  carpeting,  27  inches 
wide,  will  be  required  to  cover  a  room  15/.  9'  by  16/.  7'  ? 

629.  How  many  square  feet  in  12  boards,  averaging  12/1 
8'  long  by  I/.  9'  wide,  each  ? 

630.  What  will  it  cost  to  veneer  a  surface  7/.  6'  3"  long, 
by  5/.  2'  7"  wide,  at  S7J  cts.  per  square  foot? 

031.  How  many  cubic  feet  in  a  wall  80/.  9'  long,  I/.  8' 
wide,  and  3/.  4'  high  ? 

632.  How  many  solid  feet  in  a  pile  of  wood,  156  feet 
long,  4/.  8'  high,  6/.  4'  wide  ? 

633.  In  one  side  of  a  house  are   12  windows;  in  each 
window  12  lights:  each  light  is  ]/.  3'  by  llx.     How  many 
square  feet  of  glass  in  the  whole  ? 

13* 


150  DENOMINATE  NUMBERS.  [CHAP.  XI. 

634.  A  room  is  18/  long,  14/.  6'  wide,  9/.  8'  high.  There 
are  4  windows  in  the  room,  each  5/.  6'  long  by  3/.  wide ; 
and  2  doors,  each  6/.  9'  high  by  2/.  10'  wide.     What  will 
be  the  cost  of  plastering  said  room  at  12^-  cents  per  square 
yard  ? 

635.  What  will  it  cost  to  paint  a  house  42/.  6'  long,  28/. 
6'  wide,  19/.  6'  high,  at  13  cts.  per  square  yard  ? 

NOTE. — No  deduction  is  made  for  windows.  The  painting  of  the 
sashes  is  considered  equivalent  to  the  painting  of  the  surface  over 
which  the  sashes  stretch. 

636.  What  is  the  square  of  23/.  8'  7"?     What  is  the 
cube  of  the  same  ? 

637.  How  many  bricks,  each  8m.  long,  4m.  wide,  and 
2m.  thick,  are  required  to  build  a  wall  180  feet  long,  6/.  6' 
high,  and  three  bricks  wide,  no  allowance  being  made  for  the 
mortar  ? 

DIVISION  OF  DUODECIMALS. 

§  94.  There  are  27/  0'  7"  9"'  6""  in  the  surface  of  a  piano-cloth. 
The  breadth  of  the  cloth  is  3/.  7'  2".  What  is  its  length  ? 

3/.  7'  2")  27/.    0'     7"  9'"  6""(7/  6'  3" 
25/.     2'     2" 

If.  10'     5"  9'" 
]/.     9'     7"  0'" 


10"  9' 
10"  9' 


Dividing  the  product  27/.  0'  7"  9'"  6""  by  3/.  7'  2",  one  of  its  fac- 
tors must  give  the  other  factor. 

We  therefore  divide  first  the  27/.  by  3/.,  and  find  the  quotient  7 
feet.  Multiplying  the  whole  divisor  by  the  quotient,  we  have  25/1 
2'  2",  which  we  subtract  from  the  corresponding  denominations  of  the 
dividend.  To  the  remainder  we  annex  another  term  of  the  dividend. 


§  94.]  DUODECIMALS.  151 

Dividing  If.  10'  or  22'  by  3/.,  we  obtain  6'  for  a  quotient,  by  which 
we  multiply  the  whole  divisor.  The  product,  If.  9'  7"  0'",  we  sub- 
tract from  the  corresponding  denominations  of  the  dividend,  and  to 
the  remainder  annex  the  remaining  term  of  the  dividend.  Dividing 
10"  by  3/1  we  obtain  3"  for  a  quotient ;  multiplying  the  whole  divi- 
sor by  this,  and  subtracting,  we  find  no  remainder.  The  length  of 
the  cloth,  then,  is  7/.  6'  3". 

There  can  be  no  difficulty  as  to  how  many  indices  we  shall  annex 
to  any  term  of  the  quotient,  if  we  remember  that  the  indices  of  the 
quotient  added  to  those  of  the  divisor  must  equal  those  of  the  dim- 

0  S 

dend.     Thus,  9'""  divided  by   3"=: 


12X12X12X12X12  '  12X12 


—  or  3'"  :  so  36""""-r-6""=6"' 


12X12X12 

RULE. 

Arrange  the  numbers  as  for  denominate  division.  Di- 
vide the  highest  term  of  the  dividend  by  the  highest  term  of 
the  divisor.  Multiply  the  WHOLE  divisor  by  the  quotient 
thus  obtained,  and^  subtract  the  product  from  the  correspond- 
ing terms  of  the  dividend.  To  the  remainder  annex  the 
next  denomination  of  the  dividend.  Divide  the  highest  term 
of  this  partial  dividend  by  the  highest  term  of  the  divisor, 
as  before,  and  so  proceed,  till  the  division  is  complete. 

NOTE  1. — If,  on  the  multiplication  of  the  icholc  divisor  by  the  quo- 
tient figure,  this  is  found  too  large,  the  quotient  figure  must  be  taken 
smaller. 

2.  If  the  highest  term  of  a  partial  dividend  will  not  contain  the  di- 
visor, such  term  may  be  reduced  to  the  next  lower  denomination, 
and  the  number  in  that  denomination  added,  and  the  division  then 
performed. 

EXAMPLE'S. 

638-640.  Divide  32/.  9'  9"  by  14/.  7';  by  29/  2';  by 
7/.  3'  6". 

641.  The  area  of  a  marble  slab  is  2 1/.  1'  9" ;  its  length 
is  7/  3'.  What  is  its  breadth  ? 


152  DENOMINATE  NUMBERS.  [CHAP.  XI. 

642.  A  carpenter  bought  920  square  feet  of  boards.     He 
knew  that  their  united  length  was  480  feet.     What  did  he 
find  their  average  breadth  to  be  ? 

643.  There  were  4283cw.ft.  4'  of  earth  thrown  out  of  a 
cellar.     The  cellar  was  42/.  10'  long  and  12/6'  wide.    How 
deep  was  it  ? 

644.  I  have  a  board  fence  containing  51Qsq.ft.  10^  8". 
Its  height  is  6ft.  4m.     What  is  its  length  ? 

645.  A  block  of  marble  for  the  Washington  monument 
is  3/.  I'  wide,  2/.  3'  thick,  and  contains  37£./.  6'  11"  3'". 
What  is  its  length  ? 


ADDITION  OF  DENOMINATE  FRACTIONS. 

§  95.  We  have  seen  (§  88)  that  whole  numbers  of  differ- 
ent denominations  cannot  be  added  together ;  the  same  is 
true  of  fractions  of  different  denominate  values.  Thus,  |-  of 
a  peck  and  f  of  a  quart  cannot  be  added  together.  But  if. 
both  quantities  are  made  fractions  of  a  pccJc,  or  both  fractions 
of  a  quart,  their  sum  may  be  found. 

%pk.  =  %qt.  ;  and  %qt.+iqt.  =  6%qt.  Again,  ^qt.=-f^pJc. ; 
and  $pk.  +-fspk.=^pk.  +^P1c.  =ftpk.  =-^-qt.  =  6^qt., 
the  same  result  as  before. 

EXAMPLES. 

646-648.  Add  Is.  to  \d.  ;  \qt.  and  %pk. ;  ±da.  and  f  hr. 
649-650.  Add  \yd.  %ft.  and  $in. ;  £J  |s.  f d.  and  &qr. 
651-652.  Add  \wk.  \da.  and  ^hr.  ;  \yr.  %wlc.  ^da. 

653.  What  is  the  sum  of  -|  of  a  cwt.,  i  of  a  qr.,  J  of  a  Ib.  ? 

654.  What  is  the  sum  of  -^  of  a  bushel,  -J  of  a  peck, 
^  of  a  quart  ? 

655.  What  is  the  sum  of  T^  of  a  yard,  and  J  of  a  foot  ? 


§  96.]  DENOMINATE  FRACTIONS.  153 

656.  What  is  the  sum  of  f-  of  a  week,  £  of  a  day,  and 
£  of  an  hour  ? 

657.  What  is  the  sum  of  |  of  a  bushel,  J  of  a  peck,  and 
•1-  of  a  quart  ? 

658.  An  invalid  laborer  worked  during  the  first  week  of 
harvesting,  ^wJc.,  counting  6  days  to  the  week ;  during  the 
2d,  T7QC?a.,  counting  10  hours  to  the  day ;  during  the  3d, 
^da.  ;  during  the  4th,  6 \hr.  ;  during  the  5th,  \hr.     How 
much  did  he  earn  at  the  rate  of  12 \  cts.  per  hour? 

SUBTRACTION  OF  DENOMINATE  FRACTIONS. 

$  96.  As  in  addition,  the  fractions  must  be  first  reduced 
to  the  same  denomination,  afterwards  to  a  common  denom- 
inator. The  operation  is  then  the  same  as  subtraction  of 
common  fractions. 

From  \  of  a  pound  sterling  subtract  \  of  a  shilling. 
£l=fs.  ;  and  ^s.-ls.  =  ^s.-^s.=^s.  =  ^s.      Or, 

and  £i-£TJo  =  %Vz>-£rfo=^¥o=H*. 
the  same  result  as  before. 

EXAMPLES. 

659-660.  From  %da.  subtract  ^min.  ;  from  -f^da.  sub- 
tract ^hr.-\-^min. 

661.  From  £  of  f  of  15  yards  of  cloth  subtract  \  of  -^ 
of  1  quarter. 

662.  From  1  of  5  acres  of  land,  subtract  J  of  3  roods. 

663.  From  J  of  an  ounce,  take  f  of  a  pennyweight. 

664.  From  i  of  a  hogshead,  take  f  of  a  quart. 

665.  A  man  had  a  field  to  plough,  containing  3 A.  ^JR, 
iP. ;  |-A  f R.  |P.  was  ploughed  the  first  half-day.     How 
much  remained  to  be  ploughed  ? 


154  DENOMINATE  NUMBERS.  [CHAP.  XI. 

666.  A  grocer  lost  from  ^  of  a  hogshead  of   molasses 
\gal.  and  \qt.     How  much  of  the  hogshead,  expressed  de- 
cimally, leaked  out  ? 

667.  Suppose  a  man  consume  ^  of  every  day  in  sleep  ; 
•f-±  of  every  day  in  eating ;  %hr.  each  day  in  amusement ; 
JAr.  each  day  in  idleness  :  how  many  days,  of  10  hours  each, 
<  a^>  he,  for  work  in  the  course  of  the  year  ? 


EXERCISES  IN  DENOMINATE  FRACTIONS. 

668.  .\  person  gave  \  of  a  pound  sterling  for  a  hat,  J-  of 
a  shilling  f  >r  some  thread,  and  ^  of  a  penny  for  a  needle. 
What  did  lie  pay  for  all  ? 

669.  V7hat  is  the  value  of  -J  of  a  week,  J  of  a  day,  and 
i  of  a  minate  ? 

670.  WLat  is  the  value  of  -J  of  a  pound,  £  of  an  ounce, 
and  i  of  a  pennyweight,  Troy  ? 

671.  If  ^  pounds  of  sugar  cost  43j  cents,  how  much 
is  it  per  nound  ? 

672.  If  I  pay  $4 '04  for  8-f-  bushels  of  apples,  how  much 
do  I  give  per  bushel  ? 

673.  Four  persons,  A.,  B.,  C.,  and  D.,  own  a  ship  :  of 
which  A.  owns  ^  of  -|  of  the  whole  ;  B.  owns  |-  of  §  as  much 
as  A.  ;  C.  owns  J  as  much  as  B. ;  and  D.  owns  the  remain- 
der.    What  are  the  respective  parts  owned  by  each  ? 

674.  From  i   of  J  of  a  day  of  24  hours,  take  J  of  lj 
/iour. 

675.  To  f  of  4J  days  of  24  hours  each,  add  J  of  i  of  3j 
hours. 

676.  A  certain  sum  of  money  is  to  be  divided  between  4 
persons  in  sucn  a  manner  that  the  first  shall  have  i  of  it, 
the  second  J,  the  third  ^,  and  the  fourth  the  remainder, 
*vhich  is  $28.     What  is  the  sum  ? 


§96.]  DENOMINATE  LUMBERS.  155 

677.  A.  received  -J-  of  a  legacy,  B.  T^,  and  C.  the  remain- 
der.    Now  it  is  found  that  A.  had  $80  more  than  B.     How 
much  did  each  receive  ? 

678.  Eight  detachments  of  artillery  divided  4608  cannon- 
balls  in  the  following  manner :  the  first  took  72  and  J  of 
the  remainder  ;  the  second  took  144  and  ^  of  the  remainder ; 
the  third  took  216  and  J  of  the  remainder  ;  the  fourth  took 
288  and  £  of  the  remainder.     The  balance  was  equally  di- 
vided among  the  remaining  four  detachments.     How  many 
balls  did  each  detachment  receive  ? 

679.  Five  persons  divide  100  pounds  of  sugar  as  follows  : 
the  first  takes  ^  of  f  of  the  whole  ;  the  second  takes  i  of 
|  of  the  remainder ;  the  third  takes  J  of  -J  of  the  remain- 
der ;  the  fourth  takes  ^  of  f  of  the  remainder  ;  and  the  fifth 
had  what  was  left.     How  much  did  each  receive  ? 

080-681.  A  person  owning  100 A.  3R.  4P.  of  land,  bought 
£  of  a  farm  of  97A  lit.  30P.,  and  then  divided  J  of  the 
whole  equally  among  3  sons.  How  much  did  each  son 
have  ?  and  how  much  remained  with  the  father  ? 

682.  What  is  i  of  the  sum  of  J  of  -f-  of  13  weeks,  and 
|  of  i  of  30  days  ? 

683.  What  must  I  pay  for  38  eggs,  at  25.  2c?.  per  dozen  ? 

684.  How  much  cheese,  at  9d.  3far.  per  pound,  ought  I 
to  receive  for  13/6.  5os.  of  veal,  at  4^d.  per  pound  ? 

685.  What  is  the  value  of  J  of  a  year  of  36oi  days  -f  J 
,of  7  days  of  10  hours  each  ? 

686.  Add  |  of  4  degrees  of  69|  miles  each,  and  £  of  4 
furlongs. 

687.  If  I  buy  113/6.  13oz.  of  butter,  at  lOjc?.  per  pound, 
and  use  30/6.,  how  much  per  pound  must  I  sell  the  remain- 
der so  as  to  receive  as  much  as  the  whole  cost  ? 

688-691.  A  person  gave  J  of  all  his  money  for  a  dress 
coat,  $  of  the  remainder  for  a  pair  of  pantaloons,  and  i  of 


156  PERCENTAGE.  [CHAP.  XII. 

what  then  remained  for  a  hat.  He  then  found  that  he  had 
remaining  £3  14s.  60?.  What  was  the  cost  of  each  article? 
and  how  much  money  had  he  at  first  ? 

692.  From  a  piece  of  cloth  containing  20yd.  2qr.  2na., 
3  suits,  each  requiring  4%yd.,  were  taken ;   and  J  of  the 
remainder  was  sold  for  $10'68f .     How  much  was  that  per 
yard? 

693.  How  many  feet  in  f  of  a  statute  mile  •+-  ^  of  a 
nautical  mile  ? 

694.  How  many  inches  in  ^  of  a  yard  -f-  |  of  a  metre  ? 

695.  How  many  feet  in  J  of  a  chain  -{-  T^  of  a  furlong  ? 

696.  How  many  inches  in  ^  of  a  hand  -f-  ^  of  a  span 
4-  i  of  a  cubit  ? 

697.  How  many  inches  in  1  Ell  Scotch  -f  |  of  an  Ell 
English  +  j-  of  an  Ell  French  ? 

698.  How  many  cubic  inches  in  3  gallons,  2  quarts,  and 
1  pint  of  wine  ? 

699.  How  many  cubic  inches  in  f  of   a  gallon  -f-  %  of 
a  quart  of  beer  ? 

700.  How  many  cubic  inches  in  1  bushel  3  pecks  ? 


CHAPTER    XII. 

PERCENTAGE. 

§  97.  THE  termer  cent,  is  an  abbreviation  of  the  Latin 
words  per  centum,  which  mean  by  the  hundred.      Thus, 

2  per  cent,  signifies  2  out  of  a  hundred,  or  2  hundredths ; 

3  per  cent.  3  out  of  a  hundred,  or  3  hundredths,  &c.     Per 
cent,  is  applied  to  money,  apples,  beans,  the  pupils  of  a 
school,  or  to  any  thing  else. 


§  97.]  PERCENTAGE.  157 

We  have  seen  that  hundredths  may  be  expressed  either  as  a  com- 
mon or  as  a  decimal  fraction;  thus,  2  hundredths  =  T|6  =0*02  ; 
3  hundredths  =  T^=0'03,  <fec.  It  is  the  decimal  form  that  we  use 
in  all  operations  in  percentage  ;  0'02,  0'03,  <fec.,  are  called  the  rates 
per  cent. 

When  the  rate  per  cent,  is  more  than  100,  that  is,  more  than  1J°, 
it  is  expressed  as  an  improper  fraction  ;  or,  decimally,  as  a  mixed 
number:  thus,  106  per  cent.  =  |j{  8=1-06. 

When  the  rate  per  cent,  is  less  than  1,  that  is,  less  than  y-l^, 
it  may  be  expressed  by  a  decimal  of  three  or  four  places.  Thus 
•J  of  1  per  cent.  =  0*005  ;  which  is  read,  not  5  thousandths  per  cent., 
but  ysff  of  1  per  cent.  ;  4  of  1  per  cent.  =  0'0025  ;  which  is  read  T2/w 
of  1  per  cent.  Reduce  the  common  fraction  to  a  decimal,  writing  the 
first  quotient  figure  as  the  tenth  of  a  hundredth,  or  as  a  thousandth, 
&c. 

We  have  seen,  §  44,  that  to  obtain  a  fractional  part  of  any  num- 
ber, we  must  multiply  the  number  by  the  fraction  ;  thus,  0'06  of  25 
is  0-06X25;  0'0025  of  50  is  0'0025X50.  Hence,  to  compute  the 
percentage  of  any  number, 

Multiply  the  given  number  by  the  rate  per  cent,  expressed 
as  a  decimal.  The  product,  pointed  off  by  the  rule  for  deci- 
mals, will  be  the  percentage  required. 

EXAMPLES, 

1-8.  Express  in  figures  the  following  :  2  per  ct.  ;  8  per 
ct.  ;  12  per  ct.  ;  50  per  ct.  ;  106  per  ct.  ;  140  per  ct.  ;  260 
per  ct.  :  1800  per  ct. 

9-17.  Express  in  the  required  form 


18-25.  Express  in  decimals  ^  per  ct.  ;  J  per  ct.  ;  f  per 
ct.  ;  T9Q  per  ct.  ;  f  per  ct.  ;  f  per  ct.  ;  1  per  ct.  ;  |  per  ct. 

26-32.  Express  in  decimal  form  2|  per  ct.  ;  3j  per  ct.  ; 
5£  per  ct.  ;  36^  per  ct.  ;  220£  per  ct.  ;  500f  per  ct.  ; 
946f  per  ct. 

14 


158  PERCENTAGE.  [CHAP  XII. 

33-37.  What  is  0-05  of  $1122?  0'06  of  §79468?  0'07 
of  $8912-50?  0'08  of  $34567-62,3?  0'09  of  $3479021'05  ? 

38-47.  Find  the  percentage  of  $987654-37  at  each  of  the 
following  rates  :  J  per  ct. ;  2j  per  ct. ;  J  per  ct. ;  3  J  per  ct.  ; 
f  per  ct.  ;  6f  per  ct.  ;  25  per  ct.  ;  412  per  ct.  ;  1900 
per  ct.  ;  43  per  ct. 

48-53.  Find  38  per  ct.  of  4 ;  97  per  ct.  of  16  ;  500  per 
ct.  of  7  ;  840  per  ct.  of  28j ;  365  per  ct.  of  i  ;  92  per  ct. 


54.  What  is  3  per  cent,  of  5789  pounds? 

55.  What  is  4^  per  cent,  of  $7o'03  ? 

56.  WThat  is  7  per  cent,  of  2345  ? 

57.  What  is  30  per  cent,  of  $495  ? 

58.  A  person  laid  out  $222  as  follows:  he  gave  21  per 
cent,  of  his  money  for  calicoes  ; .  1 5  per  cent,  for  thread  ; 
45  per  cent,  for  silks ;  and  the  remaining  19  per  cent,  for 
broadcloths.     How  many  dollars  did  he  expend  for  each  ? 

59.  A  merchant  having  500  barrels  of  flour,  sold  at  one 
time  25  per  cent,  of  it ;  at  another  time  he  sold  20  per  cent, 
of  the  remainder.     How  many  barrels  did  he  sell  at  each 
time,  and  how  many  remained  ? 

60.  A  farmer  raising  1097  bushels  of  wheat,  gives  10 
per  cent,  of  it  for  thrashing ;  10  per  cent,  of  the  remainder 
for  flouring.     What  per  cent,  of  the  whole  will  he  have  left  ? 

61.  A  California  miner  having  secured  15j  pounds    of 
gold  dust,  finds  that  it  loses  5  per  cent,  in  refining ;  he  then 
gives  6  per  cent,  for  coining.     How  much  ought  he  to  re- 
ceive after  it  is  coined  ? 

62.  Suppose  at  each  stroke  of  the  piston  of  an  air-pump 
10  per  cent,  of  the  air  in  the  receiver  is  exhausted,  what 
per  cent,  of  the  air  will  remain  after  the  1st,  2d,  3d,  and 
4th  strokes,  respectively  ? 

63.  A  man  laid  out  25  per  cent,  of  $100  for  clothing  ;  at 


§  98.]  COMMISSION,  BROKERAGE.  159 

the  same  time  a  sum  of  money  was  paid  him  equal  to  25 
per  cent,  of  what  he  had  after  his  purchases.  How  much  less 
money  did  he  bring  home  than  he  had  when  he  left  home  ? 

64.  A  merchant  invested  $6480  in  trade,  but  lost  0'7o 
of  it.     How  much  did  he  save  ? 

65.  A.  and  B.  invested  $100  each  in  a  speculation.     A. 
lost  100  per  cent,  of  his  investment,  and  B.  made  100  per 
cent,  on  his.     How  much  better  off  was  B.  than  A.  ? 

66.  Out  of  a  hogshead  of  molasses  36  per  cent,  leaked. 
How  many  gallons  remained  ? 

67.  A  coal  dealer  bought  17180  tons  of  coal,  and  sold 
62  per  cent,  of  it  at  $4 '50  per  ton,  and  the  remainder  for 
$4 '8 7  per  ton.     How  much  did  he  sell  the  whole  for  ?' 

APPLICATIONS  OF  PERCENTAGE. 

§  98.  The  principle  of  percentage  has  a  very  extensive 
application  to  mercantile  transactions,  and  to  the  calcula- 
tions of  practical  life.  Commission  and  Brokerage,  Rise 
and  Fall  of  Stocks,  Assessment  of  Taxes,  Duties,  Insurance, 
Profit  and  Loss,  Interest,  Discount,  &c.,  involve  chiefly  the 
computation  of  percentage. 

COMMISSION,  BROKERAGE,  AND  STOCKS. 

§  99.  COMMISSION  is  an  allowance  made  to  an  agent  for 
the  purchase  or  sale,  or  care  of  property.  This  agent  is 
called  a  factor,  or  correspondent,  or  commission  merchant. 
Commissions  are  estimated  at  so  much  per  cent,  on  the 
money  employed. 

BROKERAGE  is  merely  the  commission  paid  to  a  Broker, 
or  dealer  in  stocks,  money,  or  bills  of  exchange,  for  trans- 
acting business. 


160  PERCENTAGE.  [cHAP.  XII. 

STOCKS  are  Government  Funds  ;  State  Bonds ;  the  capi- 
tal of  Banks  ;  of  Insurance,  Railroad,  and  Manufacturing 
Companies,  &c. 

This  capital,  or  money  paid  in,  is  divided  into  shares,  which  are 
owned  by  stockholders.  The  original  cost  of  a  share  is  its  par  value. 
If  it  sell  in  the  market  for  more  than  its  original  cost,  it  is  said  to  be 
above  par,  or  at  an  advance  ;  if  it  sell  for  less,  it  is  below  par,  or  at 
a  discount. 

The  original  cost  of  a  share  is  usually  $100,  though  it  is  sometimes 
$25,  $50,  $500,  <fcc. 

The  rise  or  fall  in  stocks  is  a  per  cent,  on  the  par  value.  Thus,  a 
share,  whose  par  value  is  -}-§£,  at  16  percent,  advance,  will  bring  1£|. 
of  its  original  cost ;  at  16  per  cent,  discount,  will  bring  but  T8^  of  its 
original  cost. 

The  profits  of  these  companies  are,  every  year,  or  every  half-year, 
divided  among  the  stockholders.  The  amount  so  paid  out  is  called 
a  dividend. 

EXAMPLES. 

68.  A  lady  had  a  bequest  of  $10500  ;  she  paid  an  agent 
2j  per  cent,  commission,  per  annum,  to  take  care  of  the 
money  for  her.     How  much  did  the  commission  amount  to  ? 

69.  I  bought  27  shares  of  Providence  Railroad  stock  at 
13  per  cent,  discount,  and  sold  them  again  at  an  advance 
of  2  per  cent.     How  much  did  I  gain  by  the  operation  ? 
The  par  value  was  $100. 

70.  A  gentleman  paid  a  broker  -J  per  cent,  to   invest 
$19278  in  government  funds.      How  much  was  the  bro- 
kerage ? 

71.  A  man  owns  46  shares  of  bank  stock,  par  value  $50, 
on  which  he  received  4  J  per  cent  dividend.     What  was  the 
amount  of  his  dividend  ? 

72.  A  factor  sells  43  bales  of  cotton,  at  $375  per  bale, 
and  charges  2  per  cent,  commission,     How  much  money 
must  he  pay  to  his  principal  ? 


§  99.]  COMMISSION,  ETC.  161 

73.  A.  sends  to  B.,  a  broker,  $3605  to  be  invested  in 
stock :  B.  is  to  receive  3  per  cent,  on  the  amount  paid  for 
the  stock.     What  was  the  value  of  the  stock  purchased  ? 

Since  B.  is  to  receive  3  per  cent.,  it  is  plain  that  $103  of  A.'s 
money  would  purchase  $100  worth  of  stock.  Hence,  the  amount  ex- 
pended for  stock  must  be  1P-J  of  $3605. 

In  such  cases  as  the  above,  when  the  given  sum  includes  the  fac- 
tor's commission,  and  we  desire  to  know  what  amount  he  must  invest 
for  his  principal  so  that  the  balance  may  be  his  commission  on  the 
amount  invested,  we  must  divide  the  given  sum  by  the  percentage 
of  the  commission  increased  by  a  unit. 

74.  A  factor  receives  $60112,  and  is  directed  to  purchase 
cotton  at  $289  per  bale  :  he  is  to  receive  4  per  cent,  on  the 
money  paid  for  the  cotton.     How  many  bales  did  he  pur- 
chase ? 

75.  The  par  value  of  125  shares  of  bank  stock  was  $100 
per  share.     What  is  the  present  value,  if  the  stock  is  worth 
1 8  per  cent,  above  par  ? 

76.  What  is  the  value  of  50  shares  of  bank  stock,  the  par 
value  of  which  was  $200  per  share,  on  the  supposition  that 
it  is  12  per  cent,  below  par,  or  that  it  is  worth  only  88  per 
cent,  of  its  par  value  ? 

77.  A  bank  fails,  and  has  in  circulation  $108567.      It 
can  pay  only  13  per  cent.     What  amount  of  money  has  it 
on  hand  ? 

78-81.  A  person  fails,  who  owes  to  A.  $3563*75,  to  B. 
$4062-35,  and  to  C.  $6723-33.  He  finds  that  he  can  pay 
65  per  cent,  of  his  debts.  How  much  ought  A.,  B.,  and  C. 
to  receive  respectively,  and  how  much  collectively  ? 

82-84.  A  person  in  trade  finds,  during  three  successive 
years,  that  at  the  end  of  each  year  his  money  has  increased 
30  per  cent.  If  he  commenced  with  $3000,  how  much 
will  he  have  at  the  close  of  each  successive  year  ? 


162  PERCENTAGE.  [CHAP.  XII. 

85.  A  tax  of  $7593-50  is  to  be  collected.     For  collecting, 
5  per  cent,  is  given,  which  must  be  collected  along  with  the 
tax.     What  was  the  whole  sura  collected  ? 

86.  Of  $1000  worth  of  gold  in  California,  5  per  cent,  is 
paid  for  transportation  to  the  United  States,  and  5  per  cent, 
of  the  balance  is  paid  for  coining.     What  is  the  value  of 
the  coin  received  ? 

87.  During  a  voyage  at  sea,  the  first  week  20  per  cent, 
of  the  provisions  are  consumed.     During  the  second  week 
40  per  cent,  of  the  balance  is  used.     What  per  cent,  of  the 
whole  remains  at  the  end  of  the  second  week  ? 

88.  How  much  is  30  per  cent,  of  30  per  cent,  of  $999  ? 

89.  A  person  purchased  350  barrels  of  flour,  and  sold 
20  per  cent,  of  it  at  $6  per  barrel.     How  much  money  did 
he  receive  ? 

90.  Of    1000  fruit-trees  4  per  cent,  die  the  first  year, 
and  5  per  cent,  of  the  remainder  die  the  second  year.     How 
many  died  during  the  two  years  ? 

\ 

ASSESSMENT  OF  TAXES. 

§  100.  TAXES  are  moneys  paid  by  the  people  for  the  sup- 
port of  government.  They  are  assessed  on  the  citizens  in 
proportion  to  their  property  ;  except  the  poll-tax,  which  is 
so  much  for  each  individual,  without  regard  to  his  property. 

Before  taxes  can  be  assessed,  an  inventory  of  all  the  taxable  prop- 
erty must  be  made. 

Next  the  sum  of  the  poll-taxes  must  be  deducted  from  the  whole 
sum  to  be  raised,  and  then  the  remainder  must  be  apportioned  ac- 
cording to  each  individual's  property. 

To  effect  this  apportionment,  find  what  per  cent,  of  the  taxable 
property  the  sum  to  be  raised  is  ;  then  multiply  each  one's  inven- 
tory by  this  per  cent,  expressed  in  decimals,  and  the  product  will  bo 
his  tax. 


§  100.] 


ASSESSMENT  OF  TAXES. 


163 


Suppose  a  tax  of  $600  is  to  be  raised  in  a  town  contain- 
ing 60  polls.  If  the  whole  taxable  property  amounts  to 
$37000,  and  each  poll-tax  is  $0'75,  what  will  be  A.'s  tax, 
whose  property  is  inventoried  at  $653,  and  who  pays  one 
poll? 


"We  find  the  sum  of  the  polls  to  be  $45 '00, 
•which  we  deduct  from  the  $600,  the  amount 
to  be  raised.  Dividing  the  remainder  $555 
by  $37000,  the  value  of  the  taxable  proper- 
ty, we  shall  have  the  tax  or  per  cent,  on  1 
dollar,  expressed  in  decimals.  Multiplying 
each  individual's  property  by  this  per  cent,  will  give  each  one's  tax. 

Having  determined  this  per  cent.,  assessors  facilitate  their  business 
by  making  out  a  table  as  follows : 


$0-75 
60 

$45-00 

555 

37000 


$600 
45 

$555 
=  $0-015 


$1  pays  $0-015 
2   "   0-03 
3   "   0-045 
4-   "   0-06 
5   "   0-075 
6   "   0-09 
7   "   0-105 
8   "   0-12 
9   "   0-135 

$10  pays  $0-15 
20   "   0-30 
30   "   0-45 
40   "   0-60 
50   "   0-75 
60   "   0-90 
70   «    1-05 
80   "    1-20 
90   "   1-35 

$100  pays  $1-50 
200   u   3-00 
300   "   4-50 
400   "   6-00 
500   "    7-50 
600   "   9-00 
700   "   1050 
800   "   12-00 
900   "   13-50 
1000   "   15-00 

The  pupil  can  easily  understand  the  application  of  the  table. 

91.  By  the  above  table,  what  would  be  the  tax  on  $425, 
there  being  no  poll-tax  ? 

92.  By  the  same  table,  what  must  B.  pay,  who  has  2 
polls,  and  whose  real  and  personal  property  is  assessed  at 
$762? 

93.  If  C.  pays  3  polls,  and  is  assessed  at  $1250,  how 
much  ought  he  to  pay  ? 

94.  What  is  the  tax  on  $375,  there  being  no  polls  ? 

95.  How  much  is  the  tax  on  $1875  ? 

96.  How  much  is  the  tax  on  $1100  ? 


164  PERCENTAGE.  [CHAP.  XII. 

97.  How  much  is  an  8  per  cent,  tax  on  an  estate  valued 
at  $17000  ? 

98.  If  a  state  tax  of  ^  of  1  per  cent,  is  levied  on  all 
the  property  of  the  state,  how  much  must  that  county  con- 
tribute whose  property  is  valued  at  $9 863473? 

99-100.  If  all  the  taxable  property  of  New  York  is  esti- 
mated at  $666089526,  what  would  be  the  whole  tax,  if 
levied  at  83  per  cent,  of  1  per  cent.  ?  What  would  the  tax 
be  at  31  per  cent,  of  1  per  cent.  ? 

101-103.  If  a  town  raise  a  tax  of  2  mills  on  the  dollar, 
how  much  must  A.  pay,  whose  property  is  estimated  at 
$10500?  How  much  must  be  paid  on  $37950?  How 
much  on  $1500  ? 

104.  I  own  real*  property  to  the  amount  of  $45650,  and 
personal*  property  to  the  amount  of  $4500.     What  will 
be  the  amount  of  my  taxes  if  my  state  tax  is  2^  per  cent. ; 
my  town  tax  3  per  cent.,  my  city  tax  4  j-  per  cent,  and  my 
school  tax  |  per  cent.  ? 

105.  What  is  the  difference  between  a  tax  of  37  per  cent, 
on  $9876,  and  a  tax  of  0 '3 7  per  cent.  ? 

CUSTOM-HOUSE  BUSINESS. 

§  101.  DUTIES  are  taxes  levied  by  government  upon  goods 
imported  into  the  country. 

These  duties,  established  by  Congress,  and  collected  by  custom- 
house officers  at  the  various  ports  of  entry,  constitute  the  revenue  of 
a  country. 

Duties  are  either  specific  or  ad  valorem.  A  specific  duty  is  a  fixed 
sum  imposed  on  a  ton,  hundred  weight,  hogshead,  gallon,  yard,  <fec., 
without  regard  to  the  value  of  the  commodity. 

*  Real  estate  or  property  consists  of  lands,  houses,  &c.,  which  cannot  be  moved. 
Personal  estate  is  property  consisting  of  stocks,  mortgages,  money,  furniture, 
goods,  &c. 


§  lUl.]  CUSTOM-HOUSE  BUSINESS.  165 

An  ad  valorem  duty  is  a  percentage  computed  on  the  cost  of  the 
article  in  the  country  from  which  it  is  imported. 

Gross  weight  is  the  entire  weight  of  merchandise,  with  the  cask, 
box,  bag,  <fcc.,  containing  it.  Net  weight  is  the  weight  of  the  mer- 
chandise after  all  deductions.  Duties  are  computed  on  the  net 
weight. 

Draft  is  an  allowance  for  waste.  Tare  is  an  allowance  for  the 
weight  of  the  cask,  box,  <fec.,  deducted  after  the  draft.  Leakage  and 
breakage  is  an  allowance  of  2  per  cent,  for  the  waste  of  liquors  in 
cask,  paying  duty  by  the  gallon ;  of  10  per  cent,  on  beer,  ale,  and 
porter  in  bottles ;  and  of  5  per  cent,  on  all  other  liquors  in  bottles. 

The  following  is  the  allowance  for  draft : 

Ib.  Ib.  Ib.  Ib. 

On  112 1  Between  336  and  1120. ..4 

Between  112  and  224  ....2  "        1120  and  2016. ..7 

"         224  and  336  ....3  Above    2016 9 

NOTE. — Though  not  mentioned  in  a  question,  draft  or  leakage  must 
be  deducted  before  the  other  specific  allowances  are  made. 

In  estimating  ad  valorem  duties,  no  deductions  of  any  kind  are  to 
te  made.  \ 

EXAMPLES. 

106.  What  is  the  duty  on  150  bags  of  coffee,  tbe  gross 
weight  of  each  bag  being  158/6.,  invoiced*  at  1  cents  per 
pound,  the  tare  being  4  per  cent,  and  duty  20  per  cent.  ? 

107.  At  40  per  cent,  ad  valorem,  what  will  be  the  duty 
on  346Z&S.  sewing- silk,  bought   at   Florence   at  $2 '50  per 
pound  ? 

108.  What  is  the  duty  on  114  barrels  of  olive  oil,  at  9 
cts.  per  gallon,  allowing  2  per  cent,  for  leakage  ? 

109.  When  there  is  a  specific  duty  on  tea  of  12  cts.  per 
pound,  what  must  be  paid  on  175  chests,  each  weighing 
112/&S.,  tare  8  per  cent.  ? 

110.  What  is  the  duty,  at  3  per  cent.,  on  47  bags  of 

*  An  invoice  is  a  schedule  of  the  articles  imported,  with  the  cost  thereof. 


166  PERCENTAGE.  [CHAP.  XII. 

pepper,  each  weighing  130  pounds  gross,  invoiced  at  5  cts. 
per  pound,  the  tare  being  3  per  cent.  ? 

111.  What  is  the  specific  duty  on  20  chests  of  tea,  at 
10  cents  per  pound,  the  gross  weight  of  the  whole  being 
378  pounds,  and  tare  on  the  whole  being  5 6lbs.  ? 

112.  What  is  the  duty,  at  3  cents  per  pound,  on  40  bags 
of  Madeira  nuts,  each  weighing  50  pounds,  the  tare  being 
3  per  cent.  ? 

113.  What  is  the  ad  valorem  duty,  at  30  per  cent.,  on 
an  invoice  of  $3400  of  broadcloths  ? 

114.  What  is  the  ad  valorem  duty,  at  55  per  cent.,  on  a 
case  of  silks  invoiced  at  $8532  ? 

115.  What  is  the  ad  valorem  duty,  at  17  per  cent.,  on 
50  bags  of  coffee,  each  weighing  97  pounds,  and  which  is 
invoiced  at  9^-  cents  per  pound  ? 

116.  What  is  the  ad  valorem  duty,  at  37  J  per  cent.,  on 
2  gross  of  cutlery,  invoiced  at  $352  ? 

117.  What  is  the  ad  valorem  duty,  at  40  per  cent.,  on 
a  case  of  silks,  invoiced  at  $3192? 

118.  What  is  the  ad*  valorem  duty,  at  50  per  cent.,  on 
a  case  of  Leghorn  hats,  invoiced  at  $1370  ? 

11'9.  What  is  the  duty  on  325  dozen . bottles  of  porter, 
at  3  cents  per  bottle,  allowance  for  breakage  being  made  ? 

120.  What  is  the  duty  on  400  dozen  bottles  of  London 
Brown  Stout,  at  4  cents  per  bottle  ? 

121.  What  is  the  duty  on  1000  bottles  of  Madeira  wine, 
at  8  cents  per  bottle  ? 

122.  What  is  the  duty  on  20  casks  of  wine,  each  con- 
taining 42  gallons,  at  13  cents  per  gallon  ? 

123.  What  is  the  duty  on  5  barrels  of  Spanish  tobacco, 
the  gross  weight  of  the  whole  being  637  pounds,  tare  5  per 
cent.,  at  5 J  cents  per  pound  ? 

124.  What  is  the  duty  on  15  hogsheads  of  molasses,  each 


§  102.]  INSURANCE.  167 

containing  63  gallons,  at  8  cents  per  gallon,  usual  allowance 
of  2  per  cent,  for  leakage  being  made  ? 

125.  What  is  the  duty,  at  3  per  cent.,  on  7  hogsheads 
of  sugar,  the  gross  weight  of  the  whole  being  9430  pounds, 
tare  being  1 5  per  cent.  ? 

126.  What  is  the  duty,  at  37  J  per  cent.,  on  a  bale  of 
Jnen,  invoiced  at  $2333  ? 

127.  What  is  the  duty,  at  15  per  cent.,  on  a  package 
of  indigo,  invoiced  at  $200  ? 

128.  What  is  the  duty,  at  30  per  cent.,  on  10  cases  of 
French  broadcloths,  invoiced  at  $8575  ? 

129.  What  is  the  duty,  at  13  cents  per  gallon,  on  35 
casks  of  wine,  each  containing  56   gallons,  usual   leakage 
being  deducted  ? 

130.  What  is  the  duty,  at  25  per  cent.,  on  a  quantity  of 
lace,  invoiced  at  $9863  ? 


INSURANCE. 

§  102.  INSURANCE  is  a  contract,  by  which  an  individual, 
or  a  company,  bind  themselves  to  make  good  any  loss  or 
damage  of  property  by  fire,  or  storms  at  sea,  or  other  cas- 
ualties. 

Ships  and  their  cargoes,  houses,  furniture,  cattle,  &c.,  are  insured. 

Life-insurance  is  a  guaranty  for  the  payment  of  a  certain  sum  of 
money  on  the  death  of  the  insured.  Health-insurance  secures  a 
weekly  allowance  during  the  sickness  of  the  insured. 

This  insurance  is  effected  in  consideration  of  a  sum  of  money, 
called  the  premium,  which  is  paid  beforehand,  to  the  insurers  or  un- 
derwriters. The  written  agreement  of  indemnity  is  called  a  policy. 

The  premium  is  estimated  at  a  certain  rate  per  cent,  on  the  amount 
insured. 


168  PERCENTAGE.  [CHAP.  XII. 

EXAMPLES. 

131.  If  A.  gets  his  house  insured  for  $1800,  at  41  cents 
on  $100,  what  will  be  the  amount  of  the  premium  ? 

132.  An  insurance  of  $12000  was  effected  on  the  ship 
Ocean,  at  a  premium  of  2  per  cent.     What  did  the  premium 
amount  to  ? 

133.  I  effected  an  insurance  of  $5230  on  my  dwelling- 
house  and  furniture  for  1  year,  at  f  of  1  per  cent.     What 
did  the  premium  amount  to  ? 

134.  What  is  the  amount  of  premium  for  insuring  $34567, 
at  60  cents  on  $100  ? 

135.  What  would  be  the  premium  for  insuring  a  ship  and 
cargo,  valued  at  $46370,  from  Boston  to  Liverpool,  at  2j 
per  cent.  ? 

136-140.  What  is  the  insurance  on  a  dwelling  and  fur- 
niture, valued  at  $15000,  at  1  per  cent.  ?  at  1^  per  cent.  ? 
at  2±  per  cent.  ?  at  2  J  per  cent.  ?  at  2  J  per  cent.  ? 

141-142.  A  house,  valued  at  $2800,  is  insured  at  45 
cents  on  $100.  What  was  the  premium?  At  48  cents  on 
$100,  what  would  have  been  the  premium  ? 

143.  A  person  at  the  age  of  38,  effects  an  insurance  on  his 
life  for  a  period  of  7  years,  for  the  sum  of  $5000,  at  the  rate  of 
$1*70  on  $100  per  annum.     What  is  the  annual  premium  ? 

144.  An  insurance  is  taken  for  a  person  aged  21  years, 
for  life  for  the  sum  of  $7500,  at  $1'82  on  $100.     What  is 
the  annual  premium  ? 

145.  A  person  at  the  age  of  18  years  effects  an  insurance 
during  life  for  the  sum  of  $10000,  at  the  rate  of  $1'69  on 
$100.     How  much  is  the  annual  premium? 

146.  A  life-insurance  for  the  sum  of  $8000  during  life, 
is  taken  by  a  person  50  years  old,  at  the  rate  of  $4*60  on 
$1 00.     What  is  the  annual  premium  ? 


§  103.]  PEOFIT  AND  LOSS.  169 

147.  A  person  going  to  California,  with  the  intention  of 
returning  at  the  end  of  3  years,  effects  an  insurance  of  $5500 
on  his  life  for  the  benefit  of  his  family,  at  1  \  per  cent,  per 
annum.  What  was  the  annual  premium  ? 

148-150.  What  is  the  annual  premium  on  a  life  insurance 
of  $18000,  at  \\  per  cent.  ?  what  at  2  per  cent.?  what  at 
2  J  per  cent.  ? 

PROFIT  AND  Loss. 

§  103.  Profit  and  Loss  signify  the  amount  which  the 
merchant  gains  or  loses  in  his  business  transactions. 

CASE    I. 

A.  bought  20  yards  of  broadcloth,  at  $1'50  per  yard, 
and  sold  the  same  at  $2*50  per  yard.  How  much  did  he 
gain?  Regained  $2'50— $r50=$l'00  per  yd. ;  $l'00x20, 
the  number  of  yards, =$20,  the  whole  gain. 

Suppose  A.  sold  the  cloth  at  $1*25  per  yard.  How  much 
did  he  lose?  He  lost  $1'50— 8l'25=$0'25  per  yard; 
$0-25X20=$5-00,  the  whole  loss. 

CASE  II. 

What  per  cent,  did  A.  gain  by  the  first  operation,  and  what 
per  cent,  did  he  lose  by  the  second  operation  ? 

Original  cost  per  yard  $1'50:  gain  per  yard  $1*00  ;  per 
cent,  gain  f.f{j  =  °'66f>  or  66f  per  cent. 

Original  cost  per  yard  $1'50  ;  loss  per  yard  $0'25  ;  per 
cent,  loss  T2/7=0'16|,  or  16|  per  cent. 

CASE    III. 

How  much  per  yard  did  A.  sell  his  broadcloth  for,  at  66f 
per  cent,  profit?  How  much  for,  at  16f  per  cent,  loss  ? 

15 


170  PERCENTAGE.  [cHAF.  XII, 

The  gain  per  yard  is  $1'50  X  0'66f  ,  which  added  to  H'oO, 
the  original  cost,  gives  for  the  selling  price  $l'50-f-$l'50  x 
0'66f,  or,  which  is  the  same  thing,  $l'50x  l'66f  =$2'50. 

Again,  the  loss  per  yard  is  3>r50xO'16j,  which  sub- 
tracted from  $1'50,  the  original  cost,  gives  for  the  selling 
price  $1-50—  $1-50  xO'16f,  or,  which  is  the  same  thing, 


CASE    IV. 

What  was  the  cost  per  yard  of  the  broadcloth,  if  A.  sold 
it  at  $2  '50  per  yard,  gaining  66|  per  cent.  ? 

What  if  he  sold  it  at  $1'25  per  yard,  losing  16f  per  cent.  ? 

The  gain  is  evidently  0*66f  of  the  original  cost,  and  $2'50, 
the  selling  price,  is  equal  to  the  original  cost,  -f  0'66f  of 
the  original  cost,  or,  what  is  the  same  thing,  equal  to  T66f 
of  original  cost  ;  hence,  the  original  cost  was  $2'50-Hl-66f 
=$1-50. 

Again,  the  loss  is  0'16|  of  the  original  cost,  and  $1'25, 
the  selling  price,  is  equal  to^the  original  cost—  0-16J  of  the 
original  cost,  or,  what  is  the  same  thing,  equal  to  0'83j  of 
original  cost  ;  hence  the  original  cost  was  8l-25-^0'83^= 
$1-50. 

NOTE.  —  The  preceding  might  be  solved  by  the  use  of  the  ratio^ 

100 

§  114.     In  the  case  of  gain  the  cost  was  —  —  -  of  the  selling  price  ; 

1065 

that  is,  $2-50  XTTT?  ;  in  the  case  of  the  loss,  the  cost  was  —  -  of  the 
1005  oof 

i  no 
selling  price,  or  $1'50  X  -r^-. 

005 

From  the  preceding  demonstrations,  we  deduce  the  fol- 
lowing 

RULES. 

I.  The  total  gain  or  loss  is  the  difference  between  the  first 
and  the  selling  price. 


§  103.]  PROFIT  AND  LOSS.  171 

II.  The  gain  or  loss  upon  a  part,  divided  by  the  cost  of 
that  part,  or  the  -whole  gain  or  loss,  divided  by  the  whole 
cost,  will  give  the  gain  or  loss  per  cent. 

III.  The  first  cost  multiplied  by  1,  plus  the  gain  per  cent.f 
or  by  1  minus  the  loss  per  cent.,  expressed  as  a  decimal,  will 
give  the  selling  price. 

IV.  The  selling  price  divided  by  1  plus  the  gain  per  cent.y 
or  by  1  minus  the  loss  per  cen{.,  expressed  as  a  decimal,  will 
give  the  cost. 

MISCELLANEOUS    EXAMPLES. 

151.  Bought  300  yards  of  broadcloth,  at  $2 -25  per  yard, 
and  sold  the  same  at  $3 '50  per  yard.       How  much  was 
gained  ? 

152.  A  merchant  bought  320  barrels  of  flour,  at  $5  per 
barrel,  but  finds  that  he  must  lose  10  per  cent,  in  the  sales. 
How  much  will  he  receive  for  the  whole  ? 

153.  Suppose  I  buy  25  cords  of  maple  wood,  at  $2'50 
per  cord,  and  sell  it  so  as  to  make  25  per  cent.     What  must 
I  receive  for  the  whole  ? 

154.  Bought  a  house  and  lot  for  $1400,  and  sold  it  for 
$1200.     How  much  per  cent,  did  I  lose  ? 

155.  Bought  225  gallons  of  molasses  for  26  cents  per 
gallon,  and  sold  the  whole  for  $6 4 '3  5.     What  did  I  gain 
per  cent.  ? 

156.  Bought  75  pounds  of  coffee,  at  10  cents  per  pound. 
At  how  much  per  pound  must  I  sell  it  so  as  to  gain  $3  on 
the  whole  ? 

157.  Bought  25  hogsheads  of  molasses,  at  $18  per  hogs- 
head, in  Havana  :  paid  duties,  $16-30  ;  freight,  $25;  cart- 
age, $5'50  ;  insurance,  $25'25.     What  per  cent,  shall  I  gain, 
if  I  sell  it  at  $28  per  hogshead  ? 


172  PERCENTAGE,  [CHAP.  XIL 

158.  If  I  buy  broadcloth  for  $3*50  per  yard,  how  much 
must  I  sell  it  at  per  yard  so  as  to  gain  25  per  cent.  ? 

159.  If  I  buy  cloth  at  $3'50  per  yard,  how  much  must 
I  sell  it  at  per  yard  so  as  to  lose  25  per  cent.  ? 

160.  A  person  bought  a  city  lot  for  $800,  and  sold  it  so 
as  to  gain  40  per  cent.     How  much  did  he  sell  it  for  ? 

161.  A  house  which  cost  $3000  was   sold  for  $2400. 
What  per  cent,  was  lost  ? 

162.  A  house  which  cost  $2400  was   sold  for  $3000. 
What  per  cent,  was  gained  ? 

163-164.  If  I  buy  an  article  for  25.  and  6c?.,  and  sell  it 
for  35.,  what  per  cent,  do  I  gain  ?  But  if  I  sell  it  for  2s., 
what  per  cent,  do  I  lose  ? 

165.  If  I  buy  eggs  at  13  cents  per  dozen,  and  sell  the 
same  at  19  cents  per  dozen,  what  per  cent,  do  I  make  ? 

166.  If  eggs  which  cost  19  cents  per  dozen,  are  sold  at 
13  cents  per  dozen,  what  is  the  loss  per  cent.  ? 

167.  I  sold  a  house  for  $4800,  on  which  I  gained  20  per 
cent.     What  did  the  house  cost  me  ? 

168.  I  bought  a  railroad  bond  for  $1055,  which  was  5^ 
per  cent,  above  par.     What  was  the  par  value  ? 

169.  Bought  50  shares  of  plank  road  stock  at  3^  per  cent, 
below  par,  for  $4825.    What  was  the  par  value  of  one  share  ? 

170.  If  I  buy  stock  at  3j  per  cent,  below  par,  and  sell 
the  same  at  5^  above,  what  per  cent,  do  I  gain  ? 

171.  If  I  buy  a  city  lot  for  $3150,  at  what  price  must 
I  sell  it  so  as  to  gain  40  per  cent.  ? 

172.  I  buy  500  barrels  of  flour,  at  $5'37J,  but  am  obliged 
to  sell  it  at  a  loss  of  15  per  cent.     What  do  I  receive  for 
the  whole  ? 

173.  A.  buys  $1000  worth  of  stock,  which  he  sells  to 
B.  at  a  gam  of  5  per  cent.     B.  in  turn  sells  the  same  to  C. 
at  a  gain  of  5  per  cent.     What  did  the  stock  cost  C.  ? 


§  IGtt.J  SIMPLE  INTEREST.  173 

174.  A.  buys  $1000  worth  of  merchandise,  which  he 
sells  to  B.  at  a  loss  of  5  per  cent.  ;  B.  in  turn  sells  the  same 
to  C.  at  a  loss  of  5  per  cent.  How  much  did  C.  give  for 
the  merchandise  ? 

175-176.  A.  buys  an  article  for  £2  3s.  6d.,  and  sells  it 
to  B.  at  a  gain  of  10  per  cent.  ;  B.  in  turn  sells  it  to  C.  at 
a  loss  of  10  per  cent.  How  much  did  C.  pay  for  the  same  ? 
What  per  cent,  of  original  cost  did  he  give  ? 

177-178.  If  I  buy  600  barrels  of  flour,  at  $5'25  per  bar- 
rel, and  sell  33^  per  cent,  of  the  same  at  a  profit  of  10  per 
cent.,  and  the  balance  at  a  profit  of  12  J  per  cent.,  how  much 
shall  I  receive  for  the  whole  ?  And  what  per  cent,  shall  I 
gain  on  the  whole  ? 

179.  Sold  a  city  lot  for  $1750,  and  find  that  I  have  lost 
12  J  per  cent.     What  did  the  lot  cost  ? 

1 80.  Sold  a  city  lot  for  $2000,  and  find  that  I  have  gained 
15  A  Per  cent-     What  did  the  lot  cost  ? 

SIMPLE  INTEREST. 

§  104.  INTEREST  is  the  sum  paid  for  the  use  of  money, 
by  the  borrower  to  the  lender.  It  is  estimated  at  a  certain 
rate  per  cent,  per  annum  •  that  is,  a  certain  number  of  dol- 
lars for  the  use  of  $100,  for  one  year.  Thus,  when  $6  is 
paid  for  the  use  of  $100,  for  one  year,  the  interest  is  said 
to  be  at  6  per  cent.  ;  when  $5  is  paid  for  the  use  of  $100 
for  one  year,  the  interest  is  said  to  be  at  5  per  cent.,  <fec. 

The  rate  per  cent,  is  generally  fixed  by  law.  In  the  New 
England  States  it  is  6  per  cent.,  while  in  the  State  of  New 
York  it  is  7  percent. 

The  sum  of  money  borrowed,  or  upon  which  the  interest 
is  computed,  is  called  the  principal.  The  principal,  with 
the  interest  added  to  it,  is  called  the  amount. 

15* 


174:  PERCENTAGE.  [CHAP.  XII. 

CASE    I. 

To  find  the  interest  on  any  given  principal,  for  any  whole 
number  of  years,  at  any  given  rate  per  cent. 

What  is  the  interest  of  $3 6 5 '50  for  3  years,  at  7  per  cent.  ? 

The  interest  of  $3 6 5 '50  for  one  year  at  7  per  cent,  is 
$365'50  X  0'07 =$25'585  ;  which  multiplied  by  3,  the  num- 
ber of  years,  gives  $76'755  for  the  interest  of  $365'50  for 
3  years  at  7  per  cent.  Hence  the  following 

RULE. 

Multiply  the  principal  by  the  rate  percent,  and  the  prod- 
uct so  obtained  by  the  number  of  years.  Point  off  as  usual. 

EXAMPLES. 

181-185.  What  is  the  interest  of  $27  for  5  years,  at  6 
percent.?  of  $98?  of  $279*50  ?  of  $33120'01  ?  of 
$6958290-035  ? 

186-190.  What  is  the  interest  of  $68  for  13  years,  at  7 
per  cent.  ?  of  $142,  for  the  same  rate  and  time  ?  of  $987*41  ? 
of  $654201*90?  of  $9412860-007? 

191-197.  Find  the  interest  on  $69582*57  for  2  years,  at 
5  per  cent. ;  at  6  per  cent.  ;  at  7  per  cent. ;  for  9  years,  at 
8  per  cent. ;  at  10  per  cent. ;  at  15  per  cent. ;  at  24  per  cent. 

198-203.  Find  the  interest  of  $9812*17  for  48  years,  at 
1 J  per  cent. ;  at  2-J-  per  cent. ;  at  3^-  per  cent. ;  at  4j  per 
cent. ;  at  5%  per  cent.  ;  at  6j  per  cent. 

204-210.  What  is  the  interest  of  $375,  at  5^  per  cent., 
for  11  years?  for  13  years?  for  15  years?  for  17  years? 
for  19  years  ?  for  23  years  ?  for  27  years  ? 

CASE  II. 

To  find  the  interest  on  any  given  principal  for  any  given 
time,  at  6  per  cent. 


§  104:.]  SIMPLE  INTEREST.  175 

The  interest  on  $1  for  one  year,  is  ^O'OG ;  and  since  2 
months  is  j2j=|-  of  a  year,  the  interest  on  $1  for  2  months 
is  80-01  ;  again,  since  6  days  is  &=-£$  of  2  months  when 
we  reckon  30  days  to  each  month,  it  follows  that  the  inter- 
est on  $1  for  6  days  is  $0-001,  Hence,  if  we  call  half  the 
number  of  months  CENTS,  and  one-sixth  the  number  of  days 
MILLS,  we  shall  obtain  the  interest  of$l  for  the  given  time,  at 
6  per  cent.  Then  the  interest  of  $1  being  multiplied  by  the 
number  of  dollars  in  the  given  principal,  will  give  the  in- 
terest sought.  As  an  example,  suppose  we  wish  the  interest 
of  $125  for  1  year,  5  months,  and  18  days,  at  6  per  cent. 

$0-085 =int.  of  $1  for  1  y.  5  m.  =  l7  months. 
3=  "          "    "    18  days, 

$0-088 =int  of  $1  for  1  y.  5  m.  and  18  days. 

If  now  we  multiply  $0'088  by  125,  the  number  of  dollars 
in  the  principal,  or,  which  is  the  same  thing,  if  we  multiply 
$125  by  0-088,  we  shall  find  $125XO'088=$11,  the  in*-er- 
est  sought. 

EXAMPLES. 

211.  What  is  the  interest  of  $49*37  for  13  months  and 
15  days,  at  6  per  cent.  ? 

212.  What  is  the  interest  of  $608'62  for  1  year  and  9 
months,  at  6  per  cent.  ? 

213.  What  is  the  interest  of  $341*13  for  7  years  and  9 
days,  at  G  per  cent.  ? 

214.  What  is  the  interest  of  $100  for  16  years  and  8 
months,  at  6  per  cent.  ? 

215.  What  is  the  interest  of  $591 '03- for  4  years,  3  months, 
and  7  days,  at  6  per  cent.  ? 

216.  What  is  the  interest  of  $0'134  for  4  months  and  3 
days,  at  6  per  cent,  ? 


176  PERCENTAGE.  [CHAP.  XIL 

217.  What  is  the  interest  of  $7 '50  for  7  months,  at  6  per 
cent.  ? 

218.  What  is  the  interest  of  $3  71 '01  for  4  years  and  15 
days,  at  6  per  cent.  ? 

219.  What  is  the  interest  of  $57*92  for  3  years,  7  months, 
and  9  days,  at  6  per  cent.  ? 

220.  What  is  the  interest  of  $329  for  5  years  and  13  days, 
at  6  per  cent.  ? 

221.  What  is  the  interest  of  $4 7 '3 9  for  1  year  and  7 
months,  at  6  per  cent.  ? 

222-224.  At  6  per  cent,  what  will  $4650  amount  to  in 
1  year  and  3  months  ?  in  2  years,  5  months,  and  10  days? 
in  2  years,  1  month,  and  6  days  ? 

225-227.  In  1  year,  6  months,  and  6  days,  at  6  per  cent., 
how  much  will  $350  amount  to  ?  How  much  will  $490 
amount  to  ?  How  much  will  $375  amount  to  ? 

228-230.  At  6  per  cent,  what  will  $1000  amount  to  in 
3  years  and  6  months  ?  in  3  years  and  6  days  ?  in  4  years 
and  4  months  ? 

CASE    III. 

To  find  the  interest  on  any  given  principal  for  any  given 
time,  at  any  given  rate  per  cent. 

First  method. 

Find  the  interest  of  $300  for  1  year,  3  months,  and  12 
days,  at  4^  per  cent. 

At  6  per  cent,  the  interest  would  be  $23*10 ;  at  4j  per  cent, 
it  would  be  4^  sixths  or  T92=}  of  $23'10=$17'325  ;  at  7 
per  cent,  the  interest  would  be  7  sixths  of  $23*10— $26'95, 
<fec.  Hence  this 

RULE. 

Find   the  interest  on  the   given   principal  for  the  given 


§  104:.]  SIMPLE   INTEREST.  1YT 

time  at  6  per  cent,  as  by  Case  II.     Then  take  as  many  SIXTHS 
of  such  interest  as  will  equal  the  given  per  cent. 

EXAMPLES. 

231.  What  is  the  interest  of  $19'41  for  1  year,  7  months, 
and  13  days,  at  7  per  cent.  ? 

232.  What  is  the  interest  of  $530  for  3   years  and  6 
months,  at  5  per  cent.  ? 

233.  What  is  the  interest  of  $5 '3 7  for  4  years  and  12 
days,  at  8  per  cent.  ? 

234.  What  is  the  interest  of  $4070  for  3  months,  at  9 
per  cent.  ? 

235.  What  is  the  interest  of  $3671  for  6  months,  at  10 
per  cent.  ? 

236.  What  is  the  interest  of  $4920'05  for  3  months,  at 

4  per  cent.  ? 

237.  What  is  the  interest  of  $40'l7  for  3  months  and  18 
days,  at  3  per  cent.  ? 

238.  What  is  the  interest  of  $3 7 '13  for  5  months  and  12 
days,  at  4^  per  cent.  ? 

239.  What  is  the  interest  of  $489  for  3  years  and  4 
months,  at  5^  per  cent.  ? 

240.  What  is  the  interest  of  $700  for  1  year  and  9  months, 
at  7  per  cent.  ? 

NOTE. — When  the  principal  is  given  in  English  money,  we  must  re- 
duce the  shillings,  pence,  and  farthings  to  the  decimal  of  a  £ ;  and 
then  proceed  as  in  Federal  Money. 

241.  What  is  the  interest  of  £75  135.  6d.  for  3  years  and 

5  months,  at  6  per  cent.  ? 

242.  What  is  the  interest  of  £14  55.  3±d.  for  4  years,  6 
months,  and  14  days,  at  7  per  cent.  ? 

243.  What  is  the  interest  of  £l  75.  Qd.  for  2  years  and 

6  months,  at  4  J  per  cent.  ? 


178  PERCENTAGE.  [CHAP.  XII. 

244.  What  is  the  interest  of  £105  10s.  6c?.  for  9^  months, 
at  5  per  cent.  ? 

245-247.  What  is  the  amount  of  $503'50  for  1  year  and 
8  months,  at  5  per  cent.  ?  what  at  6  per  cent.  ?  what  at  7 
per  cent.  ? 

248-250*  What  is  the  amount  of  8401  '13  for  3  months, 
13  days,  at  5j  per  cent.  ?  at  6j  per  cent.  ?  at  6£  per  cent.  ? 

Second  method. 

*   Find  the  interest  of  $12(J  for  3  years,  5  months,  and  15 
days,  at  7  per  cent. 

TABLE  OF  ALIQUOT  PARTS   OF  A  YEAR  OR  MONTH. 
mo.       yr. 

6   =  £  I5da.=  Joflmo. 

4   =  i-  10      =  J 

3   =  }  6       =  i 

2   =  5       =  " 


$126 
0-07 


$8-82     =1  year's  interest. 
3 

$26-46     =   3  years' 

4  mos.=£of  ayr.  2'94     =   4  months*  " 

1  mo.  =  J  of  4  mos.  735   =   1  " 

15  dys.  =4  of  1  mo.    3675  =  15  days'  " 

Ans.  $30-5025  =  3  yrs.  5  mos.  and  15  days' int. 
Hence  the  following 

RULE. 

Multiply  as  by  rule,  Case  I.     Then  find  the  interest  for 
months  and  days  by  means  of  aliquot  parts. 


§  105.]  SIMPLE  INTEREST.  179 

• 

EXAMPLES. 

251-254.  What  is  the  interest  of  $39*42  for  1  year,  5 
months,  and  11  days,  at  7  per  cent.?  of  $678*24?  of 
$9872-86?  of  $27541-03? 

255-257.  What  is  the  interest  of  $47'13  for  7  months  and 
21  days,  at  7  per  cent.  ?  at  9-|  percent.  ?  at  14 J  per  cent.  ? 

258-259.  What  is  the  interest  of  $321*21  for  3  months 
and  15  days,  at  6  per  cent.  ?  for  5  years,  9  months,  and  21 
days? 

260-262.  What  is  the  interest  of  $270  for  2  months  and 
8  days,  at  7  percent.  ?  of  $57602-01  ?  of  $4930016'02  ? 

263-265.  What  is  the  interest  of  $404'44  for  1  year, 
5  months,  and  4  days,  at  7  per  cent.  ?  of  $808*88  ?  of 
$297654-03? 

266.  What  is  the  interest  of  $99*99  for  11  months  and 
29  days,  at  5  per  cent.  ? 

267.  What  is  the  interest  of  $37*50  for  6  months  and  10 
days,  at  6J  per  cent.  ? 

268.  What  is  the  interest  of  $49*49  for  8  months  and  8 
days,  at  7  per  cent.  ? 

269-271.  What  is  the  amount  of  $4650,  at  7  per  cent., 
for  1  year  and  10  days?  for  2  years  and  3  months  ?  for  3 
years,  4  months,  and  1 2  days  ? 

272-275.  What  is  the  amount  of  $317*12  for  2  years,  5 
months,  1 8  days,  at  3  J  per  cent.  ?  at  4  per  cent.  ?  at  5  per 
cent.  ?  at  5^  per  cent.  ? 

INTEREST  WHEN  THE  TIME  is  ESTIMATED  IN  DAYS. 

§  105.  Thus  far,  we  have  considered  the  time,  for  which 
interest  is  to  be  computed,  as  estimated  in  months  and  days, 
counting  a  month  as  j^  of  a  year,  and  a  day  as  -fa  of  a 
month,  or  7fa  of  a  year.  But  as  some  months  have  31  days, 


180  PERCENTAGE.  [CHAP  XII. 

while  February  has  but  23  or  29,  we,  by  the  previous  meth- 
ods, obtain  sometimes  too  much  interest,  and  sometimes  too 
little,  though  the  error  must  always  be  small. 

There  is  a  more  accurate  method  of  computing  interest 
by  means  of  days. 

Suppose  we  wish  the  interest  of  $500  from  May  15th  to 
November  20th,  at  7  per  cent.  We  find  $500xO'07=$35 
for  one  year's  interest  of  $500,  at  7  per  cent.  By  Table 
under  §  83,  note  4,  we  find  189  days  from  May  15th  to  No- 
vember 20th. 

The  interest  for  189  days  must  be  the  same  fractional  part 
of  one  year's  interest  that  189  days  is  of  365  days.  Hence, 
$35x££f  =-**-|$p-2-  =  $18-123-}-  for  the  interest  of  $500 
from  May  15th  to  November  20th,  at  7  per  cent. 

Hence  this 

RULE. 

Find  the  interest  for  one  year.  Multiply  this  by  the 
time  expressed  in  days,  and  divide  the  product  by  365  ;  the 
quotient  will  be  the  interest  sought. 

A  note  of  $3 7 '3 7  was  given  May  3,  1848.  How  much 
was  due  on  it  Dec.  27,  1848,  at  7  per  cent.  ? 

By  the  table  under  §  83,  note  4,  we  find  238  days  from 
May  3  to  Dec.  27. 

$3 7 '3 7= principal.  365)622-5842(l'705=int.  sought. 
0-07= rate  per  cent.  365  37'37  =principal. 

$2'6159= one  year's  int.         2575     $39'075=am'nt.  Ans. 
238=time  in  days.  2555 

209272  2084 

78477  1825 
52318  259 

622-5842  •> 


106.]  PARTIAL  PAYMENTS.  181 

EXAMPLES. 

276.  A  note  of  $365  was  given  July  4,  1847.     What  will 
it  amount  to  June  1,  1849,  interest  being  7  per  cent.  ? 

277.  What  is  the  interest  on  $100  from  January  13th  to 
November  15,  it  being  leap-year,  and  interest  being  6  per 
cent.  ? 

278.  What  is  the  interest  on  $216  from  March  10th  to 
December  1st,  interest  being  5  per  cent.  ? 

279.  What  is  the  interest  on  $107  from  April  12th  to 
July  4th,  interest  being  7  per  cent.  ? 

280.  What  is  the  interest  on  $1000  from  June  20th  to 
August  13th,  interest  being  7  per  cent.  ? 

281.  What  is  the  interest  on  $730  from  July  4th  to  De- 
cember 25th,  interest  being  6  per  cent.  ? 

282.  What  is  the  interest  on  $6 3 '3 7  from  August  9th  to 
December  31st,  interest  being  7  per  cent.  ? 

283-284.  What  is  the  amount  of  $210  at  5  percent., 
from  March  1st  until  the  25th  of  the  following  December? 
What  is  the  amount  of  the  same  sum  from  July  4th  until 
January  1st,  at  7  per  cent.  ? 

285-287.  What  is  the  interest  at  5^  per  cent,  of  $325 
from  April  1st  until  August  10th?  from  August  10  until 
Oct.  5th  ?  from  Oct.  5th  until  Dec.  8th  ? 

288-290.  From  May  3d  until  August  8th,  what  is  the 
interest  on  $75,  at  5  per  cent.  ?  at  6  per  cent.  ?  at  7  per 
cent.  ? 

PARTIAL  PAYMENTS. 

§  106.  When  notes,  bonds,  or  obligations  receive  partial 
payments,  or  indorsements,*  the  rule  adopted  by  the  Su- 
preme Court  of  the  United  States  is  as  follows : 

*  From  a  Latin  phrase,  in  dor  so,  meaning  "upon  the  back  ;"  because  tho  pay 
ments  are  written  across  the  back  of  the  note. 

16 


182  PERCENTAGE.  [cHAP.  XII 

RULE. 

"  The  ride  for  casting  interest,  when  partial  payments  have 
been  made,  is  to  apply  the  payment,  in  the  first  place,  to  the 
discharge  of  the  interest  then  due.  If  the  payment  exceed 
the  interest,  the  surplus  goes  towards  discharging  the  prin- 
cipal, and  the  subsequent  interest  is  to  be  computed  on 
the  balance  of  principal  remaining  due.  If  the  payment  be 
less  than  the  interest,  the  surplus  of  interest  must  not  be 
taken  to  augment  the  principal ;  but  interest  continues  on  the 
former  principal  until  the  period  when  the  payments  taken 
together  exceed  the  interest  due,  and  then  the  surplus  is  to  be 
applied  towards  discharging  the  principal ;  and  interest  is 
to  be  computed  on  the  balance,  as  aforesaid" 

4  • 

The  above  rule  has  been  adopted  by  New  York,  Massa- 
chusetts, and  by  nearly  all  the  other  States  of  the  Union. 

CONNECTICUT    RULE. 

"  Compute  the  interest  on  the  principal  to  the  time  of  the  first  pay- 
ment ;  if  tJtat  be  one  year  or  more  from  the  time  the  interest  com- 
menced, add  it  to  the  principal,  and  deduct  the  payment  from  the  sum 
total.  If  there  be  after  payments  made,  compute  the  interest  on  the 
balance  due  to  the  next  payment,  and  then  deduct  the  payment  as 
above  ;  and  in  like  manner,  from  one  payment  to  another,  till  all  the 
payments  are  absorbed ;  provided  the  time  between  one  payment  and 
another  be  on  ?  year  or  more.  But  if  any  payments  be  made  before 
one  year's  interest  hath  accrued,  then  compute  the  interest  on  the  prin- 
cipal sum  due  on  the  obligation,  for  one  year,  add  it  to  the  principal, 
and  compute  the  interest  on  the  sum  paid,  from  the  time  it  was  paid 
up  to  the  end  of  the  year  ;  add  it  to  the  sum  paid,  and  deduct  that 
sum  from  the  principal  and  interest  added  as  above. 

"  If  any  payments  be  made  of  a  less  sum  than  the  interest  arisen  at 
ilie  time  of  such  payment,  no  interest  is  to  be  computed,  but  only  on 
^he  principal  S'^mfor  any  period" 


§  106.]  PARTIAL  PAYMENTS.  183 

$620-  UTICA,  Nov.  1,  1837. 

For  value  received,  I  promise  to  pay  Thomas  Jones,  or 
order,  the  sum  of  six  hundred  and  twenty  dollars,  on  de- 
mand, with  interest.  CHARLES  BANK. 

The  following  indorsements  -were  made  on  this  note :  1838,  Oct.  6, 
received  $61-07;  March  4,  1839,  $89'03  ;  Dec.  11,  1839,  $107-77; 
July  20,  1840,  $200'50. 

What  was  the  balance  due,  Oct.  15, 1840,  allowing  7  per  cent,  in- 
terest, according  to  the  U.  S.  rule  ? 

The  pupil  will  find  it  convenient  to  arrange  the  work  for  finding 
the  multipliers  at  6  per  cent,  as  follows : 


Date  of  note  .  .  . 

year.  mo.  da. 
...1837  10     1 

mo   da 

Multipliers 

1st  indorsement.... 
2d  indorsement 

...1838     9     6 
...1839     2     4 

11     5 

4  28 

0-055£. 
0'024f 

...1839  11  11 

9     7 

0*0464 

4th  indorsement  ... 
Date  of  settlement 

...1840.   6  20 
...1840     9  15 

7     9 
2  25 

0-0365. 
0-014£ 

35  14  0-177$. 

The  intervals  of  time  are  found  by  subtracting  the  earlier  date 
from  the  one  next  below  it,  §  89,  Ex.  471,  <fec. 

To  test  the  accuracy  of  the  work,  we  may  add  the  intervals  to- 
gether, making  35mo.  I4da. ;  and  the  multipliers  together,  making 
0'177J  Now,  subtracting  the  time  when  the  note  was  given  from 
the  time  of  settlement,  we  also  obtain  35  months  and  14  days,  which 
time  gives  0-177J  for  multiplier. 

It  is  well,  in  all  cases  where  interest  is  to  be  cast  on  a  note  of  many 
indorsements,  to  follow  the  above  method,  since  by  so  doing,  there 
is  less  chance  for  committing  errors.  In  each  particular  computation 
of  interest,  when  the  value  beyond  the  third  place  is  one-half  or 
more,  add  a  unit  to  the  decimal  in  the  third  place ;  when  that  value 
is  less  than  one-half,  neglect  it. 

Having  found  the  multipliers,  we  continue  the  work  as  follows : 


184:  PERCENTAGE.  [CHAP    XII, 

The  amount  of  note,  or  principal,  is $620-000 

Int.  on  the  same  to  Oct.  6,  1838,  at  7  per  cent.,  is 40'386 

Amount  due  on  note,  Oct.  6,  1838,  is 660'386 

The  first  indorsement  is  61*070 


599-316 
Interest  from  Oct.  6,  1838,  to  March  4,  1839,  is 17-247 

Amount  due  March  4,  1839,  is 616'563 

The  second  indorsement  is 89030 

527-533 
Interest  from  March  4,  1839,  to  Dec.  11,  1839,  is 28*414 


555-947 
The  third  indorsement  is...,  ..  107-770 


448-177 
Interest  from  Dec.  11, 1839,  to  July  20,  1840,  is 19-085 

467-262 
The  fourth  indorsement  is...  ..  200-500 


.  266-762 

Interest  from  July  20,  1 840,  to  Oct.  15,1 840,  is 4'409 

Ans.  271-171 
EXAMPLES. 

UTICA,  May  1,  1836. 

291.  For  value  received,  I  promise  to  pay  Isaac  Clark, 
or  order,  three  hundred  and  fifty  dollars,  with  interest,  at 
6  per  cent.  IS".  BROWN. 

Dec.  25,  1836,  there  was  indorsed  $50;  June  30,  1837,  $5;  Aug. 
22,  1838,  $15  ;  June  4,  1839,  $100. 

How  much  was  due  April  5,  1840,  if  interest  is  computed 
according  to  the  U.  S.  rule  ? 

292.  How  much  was  due  according  to  the  Connecticut 
rule  ? 

NOTE. — We  will  here  indicate  the  steps  of  the  process  under  the 
Connecticut  rule.      First,  find  the  amount  of  the  principal  sum  for 


§  106.]  PARTIAL  PAYMENTS.  185 

one  year;  that  is,  to  May  1,  1837.  Then  find  the  amount  of  the  first 
payment  to  the  same  date.  Deduct  the  latter  amount  from  the  for- 
mer. Next,  find  the  amount  of  the  new  principal  thus  obtained  for 
another  year,  that  is,  to  May  1,  1838  ;  then  find  the  amount  of  the 
second  payment  to  the  same  time,  and  deduct  as  before,  and  so  on. 


$1Q8T4o3o-  UTICA,  Dec.  9,  1835. 

293.  For  value  received,  I  promise  to  pay  Peter  Smith, 
or  order,  one  hundred  and  eight  dollars  and  forty-three 
cents,  on  demand,  with  interest,  at  7  per  cent. 

JOHN  SAVEALL. 

March  3,  1836,  there  was  indorsed  $50*04 ;  Dec.  10,  1836,  $13'19  ; 
May  1,  1838,  $50-11. 

How  much  remained  due,  according  to  the  U.  S.  rule, 
Oct.  9,  1840? 

294.  How  much  according  to  the  Connecticut  rule  ? 
NOTE. — After  several  steps,  there  will  be  a  new  principal,  Dec.  9, 

1838.  The  interest  is  to  be  computed  upon  this,  not  for  one  year, 
since  there  is  no  indorsement  within  the  year,  but  up  to  the  time  of 
settlement. 


{Too-  UTICA,  Aug.  1,  1837. 


295.  For  value  received,  I  promise  to  payD.  Budlong,  or 
bearer,  one  hundred  and  forty-three  dollars  and  fifty  cents, 
on  demand,  with  interest.  W.  GOULD. 

Dec.  17,  1837,  there  was  indorsed  $37'40  ;  July  1,  1838,  $7'09 ; 
Dec.  22,  1839,  $13'13;  Sept.  9, 1840,  $50'50. 

How  much  remains  due,  according  to  U.  S.  rule,  Dec.  28, 
1840,  the  interest  being  7  per  cent.  ? 

296.  How  much  according  to  Connecticut  rule  ? 

NOTE. — After  a  few  steps  we  shall  find  a  new  principal,  Aug.  1, 
1838.  We  compute  the  interest  on  this  up  to  Dec.  22, 1839,  as  there 
is  no  payment  witlu'n  a  year.  From  the  amount  deduct  the  payment 
made  Dec.  22,  1839.  We  have,  then,  another  new  principal,  the  in- 
terest on  which  is  to  be  computed  for  ono  year,  that  is,  to  Dec.  22, 

16* 


186  PERCENTAGE.  [CHAP.  XII. 

1840,  and  added ;  we  find  also  the  amount  of  the  last  payment  to 
that  date  ;  deduct,  and  find  amount  of  the  balance,  Dec.  28,  1840. 

297.  A  note  of  $486  is  dated  Sept.  7,  1831. 

March,  22,  1832,  there  was  paid  $125  ;  Nov.  29,  1832,  $150;  May 
13,  1833,  $120. 

What  was  the  balance  due,  according  to  U.  S.  rule,  April 
19,  1834,  the  interest  being  7  per  cent.  ? 

298.  What  was  due  according  to  Connecticut  rule  ? 


PROBLEMS  IN  INTEREST. 

$  107.  The  principal,  the  rate  per  cent.,  the  time,  and 
the  interest,  are  so  related  to  each  other,  that  any  three  of 
them  being  given,  the  remaining  one  can  be  found. 

PROBLEM  I. 

Given  the  principal,  the  rate  per  cent.,  and  the  time,  to 
find  the  interest, 

RULE. 

Multiply  the  interest  of  $1,  for  the  given  time  and  given 
rate  per  cent  by  the  number  of  dollars  in  the  principal. 

PROBLEM    II. 

Given  the  time,  the  rate  per  cent.,  and  the  interest,  to 
find  the  principal. 

It  is  obvious  that  the  interest  on  a  given  sum  is  as  many  times 
greater  than  the  interest  on  $1,  as  the  given  sum  is  times  greater 
than  $1.  Hence  the  following 

RULE. 

Divide  the  given  interest  by  the  interest  of  $1  for  the  given 
time  at  the  given  rate  per  cent. 


§  107.]  PROBLEMS  IN  INTEREST.  187 

EXAMPLES. 

299.  The  interest  on  a  certain  principal  for  9  months  and 
10  days,  at  4J-  per  cent.,  is  $1'01605.     What  was  the  prin- 
cipal ?  ' 

300.  What  principal  will,  in  1  year,  7  months,  and  15 
days,  at  6  per  cent.,  give  $9 '7 5  interest? 

301.  What  principal  will,  in  7  years  and  9  days,  at  6  per 
cent.,  give  $16 '86  interest  ? 

302.  What  principal  will,  in  3  years  and  6  months,  at  5 
per  cent.,  give  $92*75'  interest  ? 

303.  What  principal  will,  in  3  months  and  9  days,  at  8 
per  cent.,  give  $90  interest  ? 

304.  If  a  man's   property  be  invested  at  5j  per  cent., 
how  much  is  he  worth,  supposing  his  annual  income  to  be 
$4372-50  ? 

305.  A  widow  is  receiving  $848  per  annum.     What  is  her 
property,  supposing  it  invested  at  6  per  cent.  ? 

306.  The  annual  expenditures  of  an  orphan  asylum  are 
$2753*00.     What  fund  invested  at  7  per  cent,  will  produce 
that  amount  ? 

PROBLEM    III. 

Given  the  principal,  the  time,  and  the  interest,  to  find  the 
rate  per  cent. 

If  the  rate  per  cent,  be  doubled,  other  things  being  the  same,  the 
interest  will  be  doubled ;  if  the  rate  per  cent,  is  tripled,  the  interest 
will  be  tripled.  And,  in  all  cases,  the  interest  at  any  particular  rate 
per  cent,  is  as  many  times  greater  than  the  interest  at  1  per  cent,  as 
the  given  rate  per  cent,  is  times  greater  than  1  per  cent.  Hence  we 
have  this 

RULE. 

Divide  the  given  interest  by  the  interest  of  the  given  prin- 
cipal for  the  given  time,  at  1  per  cent. 


1&8  PERCENTAGE.  [CHAP.  XII. 

EXAMPLES. 

307.  The  interest  of  $100  for  9  months  and  10  days  is 
$3 '50.     What  is  the  rate  per  cent.  ? 

In  this  example,  we  find  the  interest  of  $100  for  9  months  and  10 
days,  at  6  per  cent.,  to  be  $4'66§.  The  interest  at  1  per  cent,  is 
$0'77£ ;  therefore,  dividing  $3'50  by  $0'77£,  we  obtain  4£  for  the 
rate  per  cent,  required. 

308.  At  what  rate  per  cent,  will  $530,  in  3  years  and  6 
months,  give  $92*75  interest? 

309.  At  what  rate  per  cent,  will  $19 -41,  in   1  year,  7 
months,  and  13  days,  give  $2'200339£  interest? 

310.  At  what  rate  per  cent,  will  $5 '3 7,  in  4  years  and  12 
days,  give  $1 '73 272  interest? 

311.  At  what  rate  per  cent,  will  $4070,  in  3  months,  give 
$91-575  interest? 

PROBLEM    IV. 

Given  the  principal,  the  rate  per  cent.,  and  the  interest, 
to  find  the  time. 

If  the  time  for  which  interest  is  computed  be  doubled,  other  things 
being  the  same,  the  interest  will  be  doubled  ;  if  the  time  is  tripled, 
the  interest  will  be  tripled.  And  in  all  cases,  the  interest  for  any 
particular  time  is  as  many  times  greater  than  the  interest  for  one 
year,  as  the  particular  time  is  greater  than  1  year.  Hence,  we  have 
this 

RULE. 

Divide  the  given  interest  by  the  interest  of  the  given  prin- 
cipal, for  1  year,  at  the  given  rate  per  cent. 

EXAMPLES. 

312.  In  what  time  will  $37' 13,  at  4j  per  cent.,  yield 
$0-7518825  interest? 


§  10  TJ  PJROBLEMS  IN  INTEREST.  189 

In  this  example,  we  find  the  interest  of  $37'13  for  1  year,  at  4£ 
per  cent,  to  be  $1-67085  ;  therefore,  dividing  $0*7518825  by  $1*67085, 
we  get  0*45  years ;  this  reduced  to  months  and  days  gives  5  months 
and  12  days. 

313.  In  what  time  will  $700,  at  7  per  cent.,  give  $85'75 
jrest  ? 

314.  In  what  time  will  $100,  at  6  per  cent.,  give  $100 
iterest  ?    That  is,  in  what  time  will  a  given  principal  double 

3lf  at  6  per  cent,  interest  ? 

315-329.  In  what  time  will  a  given  principal  double 

3lf  at  5  per  cent,  interest  ?  at  6  per  cent.  ?  at  7  ?  at  8  ? 
at  9?  at  10?  at  11  ?  at  12  ?  at  5j  per  cent.?  at  6j?  at 
71?  at  si?  a*  9^?  at  10£?  at  llj? 

330-333.  In  what  time  will  $848  amount  to  $965,  at  4 
per  cent,  interest  ?  at  5  per  cent.  ?  at  6  per  cent.  ?  at  7 
per  cent.  ? 

334-336.  A  note  for  $636*50,  at  the  time  of  its  settle- 
ment, amounted  to  $1748.  How  long  was  it  on  interest  at 
4  per  cent.  ?  at  5^  per  cent.  ?  at  6^  per  cent.  ? 

PROBLEM  V. 

Given  the  time,  rate  per  cent.,  and  amount,  to  find  the 
principal. 

This  is  the  same  as  finding  the  present  worth  of  a  debt  payable  at 
some  future  time,  without  interest;  that  is,  such  a  sum  of  money  as 
will,  if  put  at  interest,  for  the  given  time,  amount  to  the  debt. 

At  6  per  cent?  interest,  the  amount  of  $1  for  one  year  is  $1*06  ; 
therefore,  the  present  worth  of  $T06  due  one  year  hence  is  $1.  If 
the  present  worth  of  $1*06  is  $1,  the  present  worth  of  $1  will  be  the 

same  fractional  part  of  $1  that  $1  is  of  $1 -06  ;  that  is,  — —  of  $1,  or 

•       • ;  so  the  present  worth  of  $2  is  ,  <kc. 

1*1)6  1*06 


190  PERCENTAGE,  [CHAP.  XII. 

Had  the  time  been  6  months,  the  present  worth  of  81  -would  be 


At  7  per  cent,  interest,  the  present  worth  of  $1  for  one  year  would 

Si  $2 

i~oT  '  °*  ^2'  T-OT"'  ^c' 


RULE. 

Divide  the  sum,  whose  present  worth  is  required,  by  the 
amount  of  $1  for  the  given  time  at  the  given  rate  ;  the  quo- 
tient will  be  the  present  worth. 

EXAMPLES. 

337.  What  is  the  present  worth  of  $622'75,  due  3  years 
and  6  months  hence,  at  5  per  cent.  ? 

338.  What  is  the  present  worth  of  $4161-575,  due  3 
months  hence,  at  9  per  cent.  ? 

339.  What  is  the  present  worth  of  $7'10272,  due  4  years 
and  1 2  days  hence,  at  8  per  cent.  ? 

340.  Sold  goods  for  $1500,  to  be  paid  one-half  in  6 
months,  and  the  other  half  in  9  months.     What  is  the  pres- 
ent worth  of  the  goods,  interest  being  at  7  per  cent.  ? 

341.  Sold  goods  for  $1500,  to  be  paid  at  the  end  of 
7^  months.     What  is  the  present  worth  of  the  goods,  in- 
terest being  at  7  per  cent.  ? 

342.  What  is  the  present  worth  of  $50,  payable  at  the 
end  of  3  months,  at  4^  per  cent.  ? 

343.  What  is  the  present  worth  of  $3471 '20,  due  3  years 
and  9  months  hence,  at  7  per  cent.  ? 

344.  Bought  a  bill  of  goods  for  $1200,  one-third  payable 
in  3  months,  one-third  in  6  months,  and  the  remaining  one- 
third  in  9  months.     How  much  ready  cash  ought  to  pay 
for  the  goods,  if  we  consider  money  worth  6  per  cent.  ? 


§  108.]  DISCOUNT.  191 

345.  The  amount  of  a  note  due  2  years,  7  months,  and 
13  days  after  date,  is  $7l298'68.     What  is  the  principal, 
the  rate  being  6  per  cent.  ? 

DISCOUNT. 

§  108.  DISCOUNT  is  an  allowance  made  for  the  payment 
of  money  before  it  is  due.  It  is  found  by  subtracting  the 
present  worth  of  the  debt  from  the  amount  of  the  debt  at  the 
time  when  due. 

EXAMPLES, 

346.  What  is  the  discount  on  $100,  due  6  months  hence, 
at  6  per  cent.  ? 

347.  What  is  the  discount  on  $750,  due  9  months  hence, 
at  7  per  cent.  ? 

348.  What  is  the  discount  on  $150,  due  3  months  and  18 
days  hence,  at  6  per  cent.  ? 

349.  What  is  the  discount  on  $961 '13,  due  1  year  and 
5  months  hence,  at  7  per  cent.  ? 

350.  What  is  the  discount  on  $3 7 '40,  due  at  the  end  of 
7  months,  at  6  per  cent.  ? 

351-353.  Bought  a  bill  of  goods,  on  6  months'  credit, 
amounting  to  $9 7 3 '50.  How  much  ought  to  be  deducted 
if  cash  is  paid  at  the  time  of  receiving  the  goods,  interest 
being  considered  at  6  per  cent.  ?  How  much  if  interest  is 
7  per  cent.  ?  How  much  if  interest  is  8  per  cent.  ? 

354-356.  A  man  purchases  a  farm  of  97  acres,  at  $110 
per  acre,  on  a  credit  of  9  months.  How  much  would  he 
save  by  paying  cash  down  for  it,  if  interest  is  counted  at 
5  per  cent.  ?  How  much  if  it  is  estimated  at  6  per  cent.  ? 
and  how  r^uch  if  at  6j  per  cent.  ? 

357-359.  Bought  a  bill  of  goods  of  $1400,  one-half  on 


192  PERCENTAGE.  [CHAP.  XII. 

a  credit  of  6  months,  and  the  other  half  on  a  credit  of  9 
months.  If  payment  is  made  at  the  time  of  the  purchase, 
how  much  ought  to  be  deducted  if  7  per  cent,  interest  is 
considered  ?  How  much  if  5  per  cent,  is  reckoned  ?  How 
much  if  5-J  per  cent.  ? 

360-361.  A  person  at  the  age  of  18  years  has  a  legacy 
of  $500,  which  is  to  be  paid  to  him  when  he  is  21  years  of 
age.  How  much  ought  to  be  discounted  for  ready  cash, 
interest  being  6  per  cent.  ?  How  much,  interest  being  7 
per  cent.  ? 

COMPOUND  INTEREST. 

§  109.  In  making  contracts,  it  is  often  stipulated  that  the 
interest  shall  be  paid  annually  or  semi- annually,  <fec.  If  not 
paid  at  the  specified  time,  the  interest  is  added  to  the  prin- 
cipal, forming  a  new  principal,  on  which  the  next  interest  is 
to  be  computed.  The  final  amount  is  called  the  amount  at 
compound  interest.  If  from  this  the  original  principal  be 
subtracted,  the  remainder  will  be  the  compound  interest. 

What  is  the  compound  interest  of  $1000  for  3  years,  at 
7  per  cent.  ? 

Principal $1000 

Interest  on  $1000  for  one  year 70 

First  amount,  or  second  principal 1070 

Interest  on  $1070  for  one  year  74-90 

Second  amount,  or  third  principal 1144'90 

Interest  on  $1144-90  for  one  year 80'143 

Third  amount 1225'043 

Original  principal 1000 

The  compound  interest  required Ans.  $22o;043 


110.]    *  BANKING  AND  BANK  DISCOUNT.  193 


EXAMPLES. 


362.  What  is  the  amount  of  $100  at  6  per  cent,  per  an- 
num, compound  interest,  for  two  years,  the  interest  being 
payable  semi- annually  ? 

363.  What  is  the  compound  interest  of  $630  for  4  years, 
at  5  per  cent.  ? 

364.  What  is  the  amount,  at  compound  Interest,  of  $50, 
for  3  years,  at  5  per  cent.  ? 

365.  What  is  the  compound  interest  of  $1000  for  4  years, 
at  6  per  cent.  ? 

366-368.  What  will  $1700  amount  to  in  2  years,  at  6 
per  cent,  per  annum,  compound  interest,  the  interest  being 
payable  semi-annually  ?  How  much,  if  interest  is  payable 
quarterly  ?  How  much,  if  payable  annually  ? 

369-370.  What  will  be  the  compound  interest  of  $333 
for  2  years  and  6  months,  at  5  per  cent,  per  annum,  if  in- 
terest is  payable  semi-annually  ?  How  much  if  the  interest 
is  payable  quarterly  ? 

BANKING,  AND  BANK  DISCOUNT. 

§  110.  A  BANK  is  a  corporation,  chartered  by  law,  for 
the  purpose  of  receiving  deposits,  loaning  money,  dealing  in 
exchange,  and  issuing  bills  or  bank-notes,  representing  specie. 

The  money  paid  in  to  form  the  basis  for  the  business  of  a  bank  is 
called  the  capital  stock.  This  is  divided  into  shares,  and  is  owned 
by  various  individuals. 

The  affairs  of  a  bank  are  managed  by  a  board  of  directors,  chosen 
annually  by  the  stockholders.  This  board  elect  one  of  their  num- 
ber as  president, 

The  cashier,  appointed  also  by  the  directors,  superintends  the 
books,  payments,  and  receipts  of  the  bank.  He  and  the  president 
sign  all  the  bills  that  are  issued. 

The  teller  is  an  officer  who  receives  and  pays  money. 

17 


194  PERCENTAGE.  [CHAP.  XII. 

Money  is  borrowed  from  banks,  on  notes.  These  are  promises  to 
pay  a  certain  sum  at  a  specified  time.  The  person  who  signs  the 
note  is  called  the  drawer  or  maker.  The  person  to  whom  the  note 
is  made  payable,  is  called  the  payee.  A  note  to  be  negotiable,  that 
is,  to  pass  from  one  to  another  and  retain  its  value,  must  be  in- 
dorsed by  the  payee.  When  so  indorsed,  a  bank  will  discount  it ; 
that  is,  will  deduct  the  interest  from  the  amount  for  which  the  note 
is  given  (which  amount  is  called  the  face  of  the  note),  and  will  lend 
the  remainder. 

This  remainder  is  called  the  present  worth  or  proceeds. 

It  is  usual  for  the  banks  to  take  the  interest  for  3  days  more  than 
the  time  specified  in  the  note ;  and  the  borrower  is  not  obliged  to 
make  payment  till  these  three  days  have  expired,  which  are  for  this 
reason  called  days  of  grace. 

The  States  establish  the  rate  of  interest  by  law.  In  New  Eng- 
land it  is  6  per  cent. ;  in  New  York  7  per  cent.,  though  banks  in  this 
State  are  not  permitted  to  take  over  6  per  cent.,  unless  the  notes 
discounted  have  more  than  63  days  to  run. 

§111.  Bank  Discount  is  the  same  as  simple  interest  paid 
in  advance*  It  is  interest  upon  interest,  and  is,  strictly 
speaking,  usurious.  Hence,  to  compute  bank  discount, 

Cast  the  interest  on  the  face  of  the  note  for  3  days  more 
than  the  specified  time  ;  the  result  will  be  the  discount. 

The  discount  deducted  from  the  face  oj&the  note  will  give 
the  present  worth  or  proceeds  of  the  note.  ^** 

EXAMPLES. 

371.  What  is  the  bank  discount  on  $1000  for  3  months, 
at  7  per  cent.  ? 

372-376.  Find  the  bank  discount  on  each  of  the  following 
sums  for  the  time  and  at  the  rate  specified  :  $150  for  6  wo., 

*  This  method  of  taking  interest  in  advance,  being  usurious,  has  been  discon- 
tinued by  most  banks,  and  instead  thereof  they  deduct  true  discount,  as  found  by 
$108. 


§  111.]  BANK  DISCOUNT.  195 

at  6  per  cent.  ;  $375  for  3  mo.  9  da.,  at  7  per  cent.  ;  $400 
for  9  mo.,  at  7  per  cent.  ;  $2  9  '30  for  7  mo.,  at  5  per  cent.  ; 
$472  for  10  mo.,  at  7  per  cent. 

377.  A  note  for  $1800,  payable  in  60  days,  was  discounted 
at  a  bank  at  6  per  cent.     What  was  received  for  the  note  ? 

JSToiE.  —  Compute  interest  always  for  the  three  days'  grace. 

378.  What  is  the  present  worth  of  a  note  for  $6720  dis- 
counted at  a  bank,  payable  in  90  days,  at  7  per  cent.  ? 

379.  What  will  be  the  proceeds  of  a  note  for  $887'50, 
payable  in  30  days,  discounted  at  6  per  cent,  at  a  bank  ? 

380-383.  What  will  be  the  proceeds  of  the  following 
notes,  if  discounted  at  a  bank  ? 


ALBANY,  Nov.  1,  1850. 
For  value  received,  I  promise  to  pay  John  Norton,  or 
order,  six  hundred  and  fifty  dollars,  in  sixty  days  from  date, 
the  discount  being  made  at  6  per  cent. 

HORACE  ASHAM. 

$848-25.  ROCHESTER,  Aug.  3,  1850. 

Ninety  days  after  date,  I  promise  to  pay  Eli  Stetson,  or 
order,  eight  hundred  and  forty-eight  dollars  and  T2^g-,  for 
value  received,  the  discount  being  made  at  7  per  cent. 

ABRAM  MOORE. 


$69-28.  LOCKPORT,  Dec.  20,  1850. 

Four  months  from  date,  I  promise  to  pay  Enoch  Strasby, 
or  order,  sixty-nine  dollars  and  twenty-eight  cents,  value 
received,  the  discount  being  made  at  7  per  cent. 

ELLERY  STRONG. 

NOTE. — When  the  rate  of  interest  is  not  specified,  it  is  understood 
to  be  the  legal  rate  of  the  State  in  which  the  transaction  takes  place. 


196  PERCENTAGE.  [oilAP.  XII. 


84280-75.  BOSTON,  Oct.  26,  1850. 

Ninety  days  after  date,  we  promise  to  pay  Edwin  Nich- 
olson, or  order,  four  thousand  two  hundred  and  eighty  dol- 
lars and  7,  for  value  received.  NAHUM  &  WALKUM. 


$400.  UTIOA,  Feb.  1,  1851. 

Ninety  days  after  date,  I  promise  to  pay  at  the  Oneida 
Bank,  to  the  order  of  Charles  A.  Mann,  four  hundred  dol- 
lars, value  received.  JOHN  JOHNSON. 

384-385.  What  is  the  bank  discount  of  the  above  note? 
And  what  the  true  discount  as  found  by  §  108  ? 


$6QO-  ALBANY,  June  10,  1850. 

Six  months  after  date,  I  promise  to  pay  at  the  Commer- 
cial Bank,  to  the  order  of  Albertus  Williams,  six  hundred 
dollars,  value  received.  HIRON  HARTER. 


ALBANY,  June  10,  1850. 
Three  months  after  date,  I  prcjaise  to  pay  at  the  Com- 
mercial Bank,  to  the  order  of  Alberfus  Williams,  three  hun- 
dred dollars,  value  received.  HIRON  HARTER. 

386-389.  What  is  the  bank  discount  of  each  of  the  two 
foregoing  notes  ?  And  what  their  true  discounts  ? 

390.  What  would  be  the  bank  discount  of  $900,  the  sum 
of  both  notes,  for  4J  months,  the  average  time  for  which  the 
above  notes  are  made  ? 


§  112.  When  the  present  worth  of  a  bankable  note,  the 
tune  for  which  it  is  to  be  discounted,  and  the  rate  per  cent, 
are  given,  to  find  the  amount  or  face  of  the  note. 

What  must  be  the  face  of  a  bank-note  which,  when  dis- 


§  112.]  BANK  DISCOUNT.  197 

counted  for  4  months  and  15  days,  gives  a  present  worth 
of  $100,  interest  being  6  per  cent.  ? 

If  we  suppose  the  note  to  be  $1,  the  bank  discount  for  4  months 
and  15  days  will  be  $0'023  ;  hence,  $1-$0-023=$0'977,  is  the  pres- 
ent worth. 

If,  then,  $0-977  require  $1  for  the  face  of  the  note,  $200  would 
require  as  many  times  $1  as  $0'977  is  contained  times  in  $200. 
&200-7-$0-977=:$  102-35  Ans.  Hence  this 

RULE. 

Divide  the  present  worth,  or  the  amount  required  to  be 
raised.,  by  the  present  ivorth  of '$1  for  the  given  time,  and  at 
the  given  ra,te  of  bank  discount.  The  quotient  will  be  the 
face  of  the  note. 

EXAMPLES. 

391.  What  must  be  the  amount  of  a  bankable  note,  so 
that  when  discounted  for  3  months,  at  6  per  cent.,  it  shall 
give  a  present  worth  of  $600  ? 

392.  What  must  be -the  face  of  a  bankable  note,  so  that 
when  discounted  for  2  months,  at  7  per  cent.,  the  borrower 
shall  receive  $50  ? 

393.  What  must  be  the  face  of  a  bankable  note,  so  that 
when  discounted  for  10  months,  at  5  per  cent.,  the  present 
worth  may  be  $1000  ? 

394.  What  must  be  the  face  of  a  bankable  note,  so  that 
when  discounted  for  7  months,  at  7  per  cent.,  the  present 
worth  may  be  $70 '50  ? 

395.  What  amount  must  I  make  my  note,  so  that  when 
discounted  at  the  bank  for  12  months,  at  7  per  cent.,  I  may 
receive  $100? 

396.  What  must  be  the  amount  of  a  note,  so  that  when 
discounted  at  the  bank  for  6  months,  at  6  per  cent.,  the  bor- 
rower may  receive  $365  ? 

17* 


198  ANALYSIS  AND  RATIO.  [CHAP.  Xin. 

397.  A  man  bought  a  house  for  $3287  cash.  How  large 
a  note,  payable  in  90  days,  must  he  take  to  a  bank  to  real- 
ize that  amount,  at  6  per  cent,  discount  ? 

398-401.  For  what  sum  must  I  draw  my  note,  so  that 
the  bank  proceeds,  at  6  per  cent,  for  3  months,  may  be 
$150  ?  For  what  sum  that  the  proceeds  may  be  $300  ? 
For  what  sum  that  the  proceeds  may  be  $450  ?  For  what 
sum  that  the  proceeds  may  be  $500  ? 

402-403.  I  buy  a  bill  of  goods  for  $675'50,  withc-at 
credit,  and  wish  to  make  a  bank-note  for  60  days,  which, 
discounted  at  7  per  cent.,  shall  yield  this  sum.  What  must 
be  the  face  of  the  note?  What  would  be  its  face  if  dis- 
counted at  6  per  cent,  for  the  same  time  so  as  to  give  the 
same  proceeds  ? 

404.  If  a  bank-note  is  discounted  at  5  per  cent,  for  4 
months,  and  yield  $101  '75,  what  was  its  face  ? 

405-407.  I  have  three  bank-notes,  each  discounted  at  6 
per  cent. :  the  first  for  3  months,  the  second  for  4  months, 
and  the  third  for  6  months  ;  their  proceeds  were  $600,  $400, 
and  $300  respectively.  What  were  their  respective  amounts  ? 

408-410.  I  have  three  bank-notes,  each  discounted  for  6 
months :  the  first  at  5  per  cent.,  the  second  at  6  per  cent., 
and  the  third  at  7  per  cent.  They  give  equal  proceeds, 
namely,  $1000.  What  was  the  face  of  each  note  ? 


CHAPTER  XIII. 

ANALYSIS    AND    RATIO. 

ANALYSIS. 

§  113.  IF  4  men  can  do  a  piece  of  work  in  9  days,  in  how 
many  days  will  6  men  do  the  same  work  ? 


§113.]  ANALYSIS,  199 

If  4  men  can  do  the  work  in  9  days,  1  man  will  do  it  in 
4x9  =  36  days.  If  36  days  be  required  for  1  man  to  do 
the  work,  6  men  will  do  it  in  £  of  the  time ;  that  is,  in  36 
days  H-  6=:  6  days. 

This  process  of  solution  is  termed  ANALYSIS, 
NOTE. — The  word  Analysis  means  the  separating  of  any  thing  into 
its  component  parts  or  elements.  It  is  applied  here  because  the 
question  is  taken  to  pieces,  and  its  various  factors  and  their  relations 
to  each  other  are  determined  by  the  pupil's  common  sense  rather 
than  by  any  formal  rule.  It  will  be  noted  that  the  first  thing  to  be 
ascertained  in  questions  of  this  kind,  is  the  value  of  1  of  each  of  the 
unknown  quantities.  This  being  found,  the  value  required  by  the 
conditions  of  the  question  can  be  determined  by  multiplication. 

EXAMPLES, 

1-9.  If  6  men  can  dig  a  field  of  potatoes  in  1 2  days,  in 
how  many  days  will  2  men  dig  the  same  field  ?  will  3  men  ? 
4  men?  8  men?  9  men?  12  men?  18  men?  24  men? 
36  men? 

10-16.  If  12  barrels  of  cider  cost  18  dollars,  what  will 
15  barrels  cost?  18  barrels?  20  barrels?  25  barrels?  30 
barrels  ?  50  barrels  ?  100  barrels  ? 

17-22.  It  required  30  hands  8  days  to  load  a  vessel. 
How  many  days  were  required  for  6  hands  to  load  it  ?  8 
hands?  10  hands?  12  hands?  20  hands?  60  hands  ? 

23-28.  It  required  30  hands  8  days  to  load  a  vessel. 
How  many  hands  were  required  to  load  it  in  2  days  ?  hi  3 
days  ?  in  6  days  ?  10  days  ?  12  days  ?  16  days  ? 

29.  If  a  wheel  revolve  12  times  in  going  10  rods,  how 
many  times  will  it  revolve  in  going  a  mile  ? 

30.  A  man  paid  28  dollars  for  16  gallons  of  wine.     How 
much  did  he  pay  for  4  hogsheads  ? 

31.  If  20  sq.ft.  of  land  cost  $60,  what  would  be  the  price 
of  an  acre  at  that  rate  ? 


200  ANALYSIS  AND  RATIO.  [CHAP.  xm. 

32.  A  merchant  invested  in  trade  $830,  which  was  -|  of 
all  he  possessed.     What  was  his  property  ? 

NOTE. — $830=|  of  his  property  ;  |  of  it  is  S3(H-7. 

33.  A  dying  man  bequeathed  for  public  uses  8490000, 
which  was  T3F  of  his  property.     How  much  did  he  possess  ? 

34-40.  If  10  yards  of  cloth  cost  $12,  what  cost  15  yards  ? 
What  cost  20  yards?  What  cost  25  yards?  What  cost 
35  yards?  What  cost  45  yards?  What  cost  55  yards? 
What  cost  65  yards?  - 

41-44.  If  the  interest  of  $100  for  one  year  is  $6,  what 
will  be  the  interest  of  $50  for  the  same  time  ?  What  will 
be  the  interest  of  $100  for  6  months  ?  what  for  2  months  ? 
what  for  4  months  ? 

45-47.  If  4  men  mow  a  field  in  12  hours,  how  many 
must  be  employed  to  mow  it  in  3  hours  ?  How  many  to 
mow  it  in  4  hours  ?  How  many  to  mow  it  in  6  hours  ? 

48-50.  If  a  locomotive  can  run  4(^jfriles  in  one  hour,  how 
far  can  it  go  in  10  minutes  ?  How  far  in  12  minutes  ?  How 
far  in  1 5  minutes  ? 

RATIO. 

§  114.  Examples  like  the  preceding  may  be  performed  by 
RATIO. 

Ratio  is  the  relation  which  one  quantity  bears  to  another. 
It  is  expressed  as  a  quotient  arising  from  the  division  of  the 
first  quantity  by  the  second  ;  thus,  the  relation  of  7  to  8  is 
seven-eighths,  or,  expressed  as  a  ratio,  -J,  7-f-8,  7  :  8  ;  the 
latter  and  most  common  form,  being  the  division  symbol  with 
the  horizontal  line  between  the  dots  omitted. 

In  any  example,  to  obtain  the  required  value  by  the  use 
of  ratio,  it  is  simply  necessary  to  multiply  the  number  which 
is  of  the  same  denomination  as  the  value  sought,  by  the 


§115.]  RATIO.  201 

ratio  between  the  other  quantities,  which  ratio  the  conditions 
of  the  question  will  determine.     Thus, 

If  4  men  consume  20  Ibs.  of  meat  in  3  weeks,  how  many 
pounds  will  1 2  men  consume  in  the  same  time  ?  The  de- 
nomination of  the  value  sought  is  pounds  ;  hence  the  num- 
ber to  be  multiplied  is  20.  The  ratio  between  the  12  and 
the  4  is  -±f— .  Therefore  20  pounds  X  -^f—=ihe.  value  sought. 

By  cancelation,    ^  pounds  X— -=60  pounds. 

/^V  ft 

NOTE. — Every  question  must  be  carefully  examined,  to  see  wheth- 
er the  ratio  is  correctly  expressed  by  the  division  of  the  greater 
number  by  the  less,  or  of  the  less  number  by  the  greater. 

If  4  men  can  do  a  piece  of  work  in  9  days,  in  how  many 
days  will  6  men  do  the  same  work? 

The  denomination  of  the  value  sought  is  days  ;  hence  9, 
the  number  of  days,  is  to  be  multiplied. 

The  only  difficulty  now  is  to  determine  the  ratio.  The 
ratio  cannot  be  |-,  for  this  would  give  a  result  showing  that 
a  longer  time  would  be  required  for  the  6  men  to  do  the 
work  than  was  required  for  the  4  men.  The  true  ratio  is  £. 
This  may  be  proved  by  analysis.  If  4  men  do  the  work  in 
9  days,  it  will  take  one  man  4  times  as  long,  which  is  9  x  4 
days ;  and  6  men  can  do  it  in  one-sixth  of  this  time,  that 
is,  in  &J-i  days  =  9  daysx£.  In  general,  when  it  is  not 
perfectly  clear  what  the  ratio  is,  ascertain  it  by  analysis. 

§  115.  There  is  no  ratio  between  quantities  of  different 
denominations  ;  as,  for  example,  between  2  yards  and  4  feet, 
between  7  dollars  and  13  cents,  or  between  £3  4s.  and  3s.  4d. 
Before  the  ratio  of  two  quantities,  whether  consisting  of  one 
or  more  denominations,  can  be  determined,  they  must  be  re- 
duced to  the  same  lowest  denomination.  2  yards =6  feet; 


202  ANALYSIS  AND  KATIO.  [CHAP.  XIII. 

$7  =  700  cents  ;  £3  4s.  =  768c/.  ;  3s.  4d.  =  40d.     Hence,  tho. 
ratios  of  the  preceding  quantities  are  -|-  ; 


EXAMPLES. 

51-55.  What  is  the  ratio  of  6  inches  to  9  inches?  to  2 
feet  ?  to  6  feet  ?  to  3  rods  ?  to  1  mile  ? 

56-60.  What  is  the  ratio  of  7  pence  to  8  shillings  ?  to 
9s.  6d.  ?  to  18  shillings  ?  to  £3  ?  to  £2  2s.  2d.  2?/.  ? 

61.  What  part  of  50  men  is  2  men  ?  that  is,  what  is  the 
ratio  of  2  men  to  50  men  ? 

62-65.  What  is  the  ratio  of  7  per  cent,  to  8  per  cent.  ? 
to  4i  per  cent.  ?  to  5^  per  cent.  ?  to  7J  per  cent.  ? 

66.  What  part  of  3  miles,  40  rods,  is  27  feet,  9  inches  ? 
that  is,  what  is  the  ratio  of  27/3f.  9m.  to  3m.  40rd.  ? 

67.  What  part  of  1  day,  9kr.  is  17  minutes,  4  seconds? 

68.  What  part  of  $700  is  $5'30  ? 

69.  What  part  of  2  hogsheads  is  3  pints  ? 

70.  What  part  of  $3  is  2|  cents  ? 

71.  What  part  of  10  shillings,  8   pence,  is  3  shillings, 
one  penny  ? 

72.  What  part  of  100  acres  is  63  acres,  2  roods,  7  rods 
of  land  ? 

73.  In  the  Eagle  there  are  232  J  grains  of  pure  gold,  and 
12T9Q-  grains  of    silver,  and  the  same  quantity  of  copper. 
The  silver  and  copper  is  each  what  part,  by  weight,  of  the 
gold  ?     And  the  silver  and  copper  together  is  what  part 
of  the  gold  ? 

74.  In  the  United  States  standard  silver  coin  of  one  dol- 
lar, there  are  37  lj  grains  of  pure  silver,  and  41-J  grains  of 
copper.     What  fractional  part  is  the  copper  of  the  silver  ? 

75.  The  silver  in  standard  gold  coin  is  what  part  of  the 
silver  in  the  same  value  of  standard  silver  coin  ? 

76.  The  pound  Troy  contains  5760   grains,  the  pound 


§  115.]  RATIO.  203 

Avoirdupois  contains  7000  grains.     A  pound  Troy  is  what 
part  of  a  pound  Avoirdupois  ? 

77.  The  Imperial  gallon  contains  277-J  cubic  inches,  near- 
ly ;  the  old  wine  gallon  contains  231.     What  part  of  the 
Imperial  gallon  is  the  old  wine  gallon  ? 

78.  The  solar  year  is  365  days,  5  hours,  48  minutes,  48 
seconds.     By  what  part  of  a  day  does  this  exceed  365  days  ? 


PROMISCUOUS    EXAMPLES. 

79-83.  If  a  ship's  crew  of  30  men  consume  $900  worth 
of  provisions  during  a  voyage  of  60  days,  how  many  dollars 
worth  would  they  consume  during  a  voyage  of  1 1 7  days  ? 
of  30  days  ?  of  45  days  ?  of  72  days  ?  of  99  days  ? 

84-88.  If  a  ship's  crew  of  30  men,  during  a  voyage  of 
60  days,  consume  $900  worth  of  provisions,  how  many  dol- 
lars worth  would  a  crew  of  7  men  consume  in  the  same 
time?  how  many  would  15  men?  45  men?  72  men?  117 
men? 

89.  If  f  of  a  man's  furniture  be  worth  860  dollars,  what 
is  the  whole  of  it  worth  ? 

90.  Five  men  earn  $32  in  a  week.     How  much  can  84 
men  earn  in  the  same  time  ? 

91.  Five  men  earn  $32  in  a  week.      How  many  weeks 
will  it  take  20  men  to  earn  the  same  sum t? 

92-94.  Five  men  earn  $32  in  a  week.  How  many  men 
will  it  take  to  earn  the  same  sum  in  a  day,  or  one-sixth  part 
of  a  week  ?  in  an  hour,  allowing  10  working  hours  hi  a  day  ? 
in  a  minute  ? 

95.  If  a  post  4  feet  high  cast  a  shadow  of  12  feet,  how 
long  will  that  pole  be  that  casts  a  shadow  of  240  feet  ? 

96.  If  a  post  4  feet  high  cast  a  shadow  of  1 2  feet,  how 
long  a  shadow  will  a  pole  90  feet  high  cast  ? 


204  ANALYSIS  AND  EATIO.  [CHAP.  XIII. 

97.  How  high  is  yonder  steeple  ?  Its  shadow  is  14  feet 
long ;  and  the  shadow  of  this  fence,  4j  feet  high,  is  8  inches. 
-  98.  Suppose  a  poll-tax  of  $1344  be  laid  on  a  town  con- 
taining 4370  inhabitants.  What  proportional  tax  must  be 
laid  on  a  town  containing  721  inhabitants? 

99.  It  took  140  laborers  9  days  to  dig  a  canal  14  rods 
long.  How  long  would  it  take  the  same  laborers  to  dig  a 
canal  Im.  17n£.  2yd.  2ft.  long? 

100-106.  If  one  gross  of  lead  pencils  is  worth  $1  50,  what 
is  1  dozen  worth?  2  dozen?  3J  dozen?  4j  dozen?  5f 
dozen  ?  7  J  dozen  ?  9  dozen  ? 

107-112.  If  it  require  84  bushels  of  apples  to  make  8 
barrels  of  cider,  how  many  barrels  will  42  bushels  make  ? 
How  many  will  100  bushels  make?  How  manvjrfll  168 
bushels  make  ?  How  many  bushels  will  be  required  to 
make  1  barrel  ?  How  many  to  make  6  barrels  ?  How 
many  to  make  10  barrels  ? 

113-121.  If  there  are  165  feet  in  10  rods,  how  many  feet 
are  there  in  2  rods  ?  in  4  rods  ?  in  6  rods  ?  in  15  rods  ?  in 
20  rods  ?  in  25  rods  ?  How  many  rods  in  330  feet  ?  How 
many  rods  in  66  feet  ?  How  many  rods  in  132  feet  ? 

122-126.  If  5  men  can  reap  a  field  of  grain  in  3  days  of 
10  hours  each,  how  many  men  would  reap  it  in  10  hours  ? 
How  many  in  15  hours?  How  many  in  25  hours  ?  How 
many  in  6  hours  ?  How  many  in  30  hours  ? 

127-137.  If  $100  gain  $7  interest  in  12  months,  how 
much  will  it  gain  in  3  months  ?  how  much  in  5  months  ?  in 
7  months?  in  11  months?  in  17  months?  How  much  will 
$300  gain  in  12  months  ?  How  much  will  $500  gain  in  12 
months?  How  much  will  $700  gain  in  12  months?  How 
much  will  be  required  to  gain  $10  in  12  months?  How 
much  to  gain  $18  in  same  time  ?  How>much  to  gain  $24  ? 

138-141.  If  -I  of  a  ship  is  worth  $49000,  what  is  the. 


.16.]  PRACTICE.  205 

lole  worth?  what  is  ^  of  it  worth?  what  is  f  \\orth? 
rhat  is  4  worth  ? 

o 

142-150.  If  a  person  can  count  300  in  one  minute,  how 
long  will  he  require  to  count  45  ?  how  long  to  count  75  ? 
how  long  to  count  225  ?  How  many  can  he  count  in  5  sec- 
onds ?  how  many  in  13  seconds  ?  how  many  in  50  seconds  ? 
how  many  in  75  seconds?  how  many  in  17  seconds  ?  how 
many  in  37  seconds  ? 

PRACTICE. 

§  116.  PRACTICE  is  the  employment  of  the  ratio  of  a  mul- 
tiplier or  divisor  to  its  unit  of  the  same  kind,  instead  of  the 
employment  of  the  given  multiplier  or  divisor  itself.  Thus, 
if  a  bushel  of  apples  be  worth  50  cents,  what  will  18^-  bush- 
els be  worth  ?  This  answer  may  be  obtained  by  multiply- 
ing 50  cents  by  18^-  giving  $9 '25.  But  the  ratio  of  50  cents 
to  $1.00  is  Y£O  =  J.  Therefore,  to  find  how  many  dollars 
1 8 J  bushels  are  worth,  it  is  only  necessary  to  multiply  1 8  J, 
the  number  of  bushels,  by  \ ;  that  is,  to  divide  18£  by  2. 

What  is  the  interest  of  $740  for  a  year  and  6  months  ? 

For  a  year  the  interest  is  $740  x  0'06 =$44-40.  Now  as 
6  months  is  \  of  a  year,  we  find  the  interest  for  6  months, 
by  taking  \  of  $44'40,  which  is  $22'20.  Hence  the  interest 
for  one  year  and  6  months  is  $44'40-f  $2^-20  =  $66'60. 

For  tables  of  the  ratios  of  particular  parts  of  a  dollar  to 
their  unit,  see  §  70,  and  of  fractional  parts  of  a  year  or  month, 
to  their  units  respectively,  see  §  104,  Case  III.,  Second 
method. 

EXAMPLES. 

151.  What  will  435  yards  of  cloth  cost,  at  $0'7o  per  yard  ? 

152.  If  I  receive  7  dollars  for  the  use  of  $100  for  one 

18 


206  ANALYSIS  AND  KATIO.  [CHAP.  XIII 

year,  how  much  ought  I  to  receive  for  the  use  of  $100  for 
7  months  and  1 8  days  ? 

153.  What  cost  7-J  cords  of  wood,  at  $2 '75  per  cord  ? 

154.  What  is  the  value  of  28f  pounds  of  butter,  at  11 
cents  per  pound  ? 

155.  What  is  the  value  of  500  J  yards  of  tape,  at  2j  cents 
per  yard  ? 

156.  What  must  I  give  for  13f  bushels  of  oats,  at  43| 
cents  per  bushel  ? 

157.  What  cost  18f  pounds  of  ham,  at  8  cents  per  pound  ? 

158.  What  cost  15f  gallons  of  oil,  at  SO'75  per  gallon? 

159.  What  cost  4000  quills,  at  $>2'25  per  1000  ? 

160.  What  cost  27|  yards  of  carpeting,  at  87  J  cents  per 
yard? 

161.  What  is  the  value  of  25  bushels  of  potatoes,  at 
$0-3  lj  per  bushel? 

162.  What  is  the  value  of  54  spelling-books,  at  12-J  cents 
per  copy  ? 

163.  What  is  the  value  of  47J  reams  of  paper,  at  $3'25 
per  ream  ? 

164.  What  is  the  value  of  30£  gross  of  almanacs,  at  $2 '2 5 
per  gross  ? 

165.  What  cost  16|  gallons  of  vinegar,  at  16-|  cents  per 
gallon  ? 

166.  What  is  the  value  of  5^  bushels  of  walnuts,  at  $1-62^ 
per  bushel  ? 

167.  What  cost    3|  gross  of   matches,  at  $1'125   per 
gross  ? 

168.  What  cost  325  bushels  of  apples,  at  37J  cents  per 
bushel  ? 

169.  What  cost  16J  yards  of  cloth,  at  $3j  per  yard  ? 

170.  If  the  interest  on  a  certain  sum  of  money  is  $7*35 
in  one  year,  how  much  will  it  be  for  5 \  months  ? 


§  117.]  REDUCTION  OF  CURRENCIES.  207 

171.  If  the  interest  of  $100  for  one  year  is  $6,  how  much 
is  it  for  10  months  and  10  days? 

172.  If  a  steam  locomotive  pass  18  miles  in  1  hour,  how 
far  will  it  move  in  50^  minutes  ? 

173.  If  the  interest  of  $100  for  12  months  is  $7,  how 
much  is  it  for  4 J  months  ? 

174.  What  must  I  pay  for  l£  cords  of  wood,  128  feet  in 
a  cord,  at  6j  cents  per  foot  ? 

175-180.  For  $360  how  many  bushels  of  apples  can  I 
buy  at  50  cents  per  bushel?  how  many  at  25  cents  per 
bushel  ?  how  many  at  12£  cents  ?  how  many  at  33  J  cents  ? 
how  many  at  20  cents  ?  how  many  at  16 J  cents? 

181-185.  Among  how  many  beggars  can  $12  be  distrib- 
uted by  giving  6J  cents  to  each  ?  how  many  if  each  receive 
10  cents  ?  how  many  if  each  receive  12^  cents  ?  how  many 
if  each  receive  16  j-  cents  ?  how  many  if  each  receive  20  cents  ? 

186-190.  If  my  income  is  $600  per  annum,  how  much 
will  it  be  for  3  months  ?  how  much  for  15  days  ?  how  much 
for  10  days  ?  how  much  for  5  days  ?  how  much  for  1  day  ? 

191-195.  How  many  yards  of  carpeting  can  be  bought 
for  $300  at  $1'12£  per  yard  ?  how  many  at  $1'25  per  yard  ? 
how  many  at  $T87i  per  yard  ?  how  many  at  $2'06-J-  per 
yard  ?  how  many  at  $2'16f  ? 

196-200.  If  the  earth  move  68000  miles  per  hour  hi  its 
orbit,  how  far  will  it  move  in  35  minutes  ?  how  far  in  45 
minutes  ?  how  far  in  55  minutes  ?  how  far  in  Ihr.  35min.  ? 
how  far  in  1h.  10mm.  ? 

REDUCTION  OF  CURRENCIES. 

§  117.  Currency  is  money,  whether  specie,  consisting  of 
domestic  and  foreign  coins,  or  bank-notes,  redeemable  in 
specie. 


208  ANALYSIS  AND  RATIO.  [CHAP.  XIII, 

Foreign  coins  have,  first,  an  intrinsic  value,  determined 
by  their  weight  and  purity  ;  secondly,  a  commercial  value, 
which  is  the  price  they  will  bring  in  the  market  ;  thirdly,  a 
legal  value,  which  is  the  value  established  by  law. 

Thus,  the  Pound  Sterling  (English)  is  represented  by  a 
gold  coin  called  a  sovereign.  Its  intrinsic  value,  as  com- 
pared with  our  gold  eagle  of  latest  coinage,  is  $4'861.  Its 
commercial  value  depends  upon  the  state  of  trade  between 
this  country  and  England.  If  the  balance  of  trade  be 
against  us,  requiring  the  transportation  of  coin  to  pay  our 
debts,  the  sovereign  will  command  a  higher  price  than  if  we 
owe  nothing  abroad,  and  consequently  require  no  specie  for 
shipment.  This  mercantile  value  varies  from  $4  '83  to  $4'86. 

The  legal  or  custom-house  value  of  the  sovereign  is  $4-84, 
as  fixed  by  act  of  Congress  in  1842/ 

§  118.  To  reduce  Sterling  to  Federal  Money. 

First  method.  £lz=$4'84;  consequently,  multiplying 
$4*84  by  the  number  of  pounds,  will  give  their  value  in 
Federal  Money. 

NOTE.  —  If  there  arc  shillings,  pence,  or  farthings  in  the  given  quan- 
tity, they  must  he  reduced  to  the  decimal  of  a  pound  before  multi- 
plication. 

Example.  What  is  the  value  of  £9  5s.  in  Federal  Money  ? 
£9  5s.  =  £9'25 


§  119.  To  reduce  Federal  to  Sterling  Money. 

$4'84  =  £l  ;  consequently,  dividing  the  given  number  of 
dollars  by  4'84,  the  number  of  dollars  in  a  £,  will  give  a 
quotient  in  pounds  and  the  decimal  of  a  pound.  The  deci- 
mal must  be  reduced  to  its  equivalent  value  in  shillings, 
pence,  and  farthings. 


§  120.]        REDUCTION  OF  CURRENCIES.  209 

Example.  Reduce  $44*77  to  its  value  in  Sterling  Money. 
44'77-f-4"84:=9<25,  the  number  of  pounds  sterling  =  £9  5s. 

§  120.  Method  by  ratio.  There  is  another  mode  of  per- 
forming these  reductions,  which  is  a  more  accurate  mercan- 
tile method. 

The  original  value  of  the  pound  sterling,  as  fixed  by  act 
of  Congress  in  1799,  was  $4f  or  $4'444  +  .  This  value  is 
called  the  par  value  of  £l ;  but  it  is  now  too  small  by  a  va- 
riable percentage  of  itself.  Consequently  this  percentage, 
called  the  premium  of  exchange,  must  be  added  to  the  par 
value  to  give  the  current  mercantile  value  of  the  pound. 

Thus,  suppose  exchange  on  England  is  at  9  per  cent,  pre- 
mium, £l=$4^  x  l'09=par  value  of  £l  plus  the  premium 
of  exchange  ;  if  exchange  be  at  10  per  cent,  premium, 
£l=$4£xl-10,  &c.  So  conversely,  $l  =  £T9o^-l'09,  &c. 

Example.  Reduce  £9  5s.  to  Federal  Money,  when  the 
premium  of  exchange  is  9  per  cent. 

£9  5s.  =  £9'2o  ;  $4£x9-25=$41-lll  +  .  And  $41-111 
Xl-09=$44-81099-f  Ans. 

Reduce  $44'81099-f  to  British  currency. 

44-81099-^4^=44-81099  XT9o  =  10'08249-f,  the  num- 
ber of  pounds  at  par  value.  £lO-OS249-f-l'09  =  £9-25 
nearly.  So  that  £9  5s.  is  the  answer. 

Hence  to  change  Sterling  to  Federal  Money, 

Reduce  the  pounds  and  decimal  of  a  pound  at  their  par 
value  to  dollars ;  then  multiply  the  result  by  a  percentage 
that  will  express  the  par  value  plus  the  premium  of  exchange. 

To  change  Federal  to  Sterling  Money. 

Reduce  the  dollars  and  decimal  of  a  dollar  to  pounds  at 
their  par  value  ;  then  divide  the  result  by  a  percentage  that 
will  express  the  par  value  plus  the  premium  of  exchange. 

18* 


210  ANALYSIS  AND  KATIO.  [CHAP.  XIJI. 

§  121.  Many  of  the  States,  at  the  present  day,  make  use 
of  the  denominations  of  Sterling  Money  to  some  extent. 
But  the  value  of  the  pound  and  its  parts,  as  will  be  seen  by 
the  table,  is  not  the  same  in  all  the  States.  (For  the  rea- 
son of  this,  see  §  70,  Note.) 


TABLE. 

$1  'm 


{   Georgia^01^'  [=45.  8fc£^,  caUed  Georgia  currency 
"  'm  {  Scotia,   |  =  5,=£i  called  Canada  cm-rency. 

.r  New  England  States,   -^ 
SI  in  J   ^guila,  1=6*.^^,  called  New  England 

1   Kentucky,  currency. 


L  Tennessee, 

r  New  Jersey,      ^ 

Pennsylvania,       =  Vs.  6<£=£f,  called  Pennsylvania  cur- 

$1  in  -{   _  .     J  ^ 

Delaware,  rency. 

I  Maryland,          J 
(  New  York,         ) 

$1  in  •<  Ohio,  f  ==  8s.=£|,  called  New  York  currency. 

(  North  Carolina,  ) 

We  have,  by  the  table,  the  value  of  $1,  expressed  as  the 
fraction  of  a  pound  in  the  various  currencies.  It  is  obvious 
that  by  inverting  the  ratio  expressed  by  those  fractions,  we 
shall  obtain  the  value  of  £1,  of  each  of  the  above  currencies, 
in  the  fraction  of  a  dollar. 

Hence,  to  reduce  Federal  Money  to  Canada  or  to  any 
State  currency, 

Multiply  the  sum  in  Federal  Money  by  the  value  of  $1 
expressed  as  the  fraction  of  a  pound  of  the  currency  to  which 
the  sum  is  to  be  reduced.  If  the  product  contain  the  decimal 
of  a  pound,  reduce  it  to  shillings  and  pence. 


§  122.]  REDUCTION  OF  CURRENCIES.  211 

To  reduce  Canada  or  any  State  currency  to  Federal  Money, 
Multiply  the  given  sum,  reduced  to  pounds  and  the  deci- 
mal of  a  pound,  by  the  value  of  £l  of  the  given  currency, 
expressed  as  the  fraction  of  a  dollar. 

§  122.  A  table  of  some  of  the  foreign  coins  at  tlieir  cus- 
tom-house value. 

Pound  Sterling  or  Sovereign $4'84 

Guinea,  English  5*00 

Crown,      "           1-06 

Shilling  piece,  English -23 

Louis-d'or,  French  4'56 

Franc,             "        -186 

Doubloon,  Mexico 15'60 

Silver  Rouble  of  Russia 0'75 

Florin  or  Guilder  of  the  United  Netherlands 0'40 

Mark  Banco  of  Hamburg 0'85 

Real  of  Plate  of  Spain 0-10 

Real  of  Vellon  of  do 0'05 

Milree  of  Portugal 112£ 

Tale  of  China T48 

Pagoda  of  India 1-84 

Rupee  of  Bengal 0'50 

Specie  dollar  of  Sweden  and  Norway 1-06 

Specie  dollar  of  Denmark T05 

Thaler  of  Prussia  and  N.  States  of  Germany 0'69 

Florin  of  Austrian  Empire  and  City  of  Augsburg,  0'48£ 

Lira  Lombardo- Venetian  Kingdom  and  of  Tuscany,  0'16 

Ducat  of  Naples 0'80 

Ounce  of  Sicily 2'40 

Pound  of  British   Provinces,   Nova  Scotia,  New 

Brunswick,  Newfoundland,  and  Canada  4'00 

Rix-dollar  of  Bremen 0'78£ 

Thaler  of  Bremen  0'7l 

Mil-rees  of  Madeira TOO 

"          of  Azores 0'83£ 

Rupee  of  British  India 0'4H 

10  Thalers,  German ,  7'80 


212  ANALYSIS  AND  RATIO.  [CHAP.  XIII. 

Foreign  coins  may  obviously  be  reduced  to  Federal  Money,  by 
multiplying  the  United  States  value  of  one  coin  by  the  number  of 
coins.  Federal  Money  may  be  reduced  to  its  value  in  a  required 
foreign  coin,  by  dividing  the  given  sum  of  money  by  the  value  of  on* 
such  coin  expressed  in  Federal  Money. 


PROMISCUOUS    EXERCISES    IN    REDUCTION    OF    CURRENCIES. 

201-210.  Reduce  the  following  sums,  U.  S.  currency,  to 
Sterling  Money,  at  custom-house  value:  $4'84  ;  $19"605; 
$32-48;  $59-00;  $876'49;  $27'18  ;  $1264-36  ;  $22096'27  ; 
$446987-84;  $2768912-76. 

211-218.  When  the  premium  of  exchange  on  England 
is  9  per  cent.,  what  is  the  value  of  the  following  sums  in 
British  currency?  $8'72 ;  $24'986 ;  $79'484;  $712'45 ; 
$8694-36;  $79823'12j;  $8942T07  ;  $216549'48. 

219-234.  What  is  the  value  of  the  preceding  sums,  in 
British  currency,  when  the  premium  of  exchange  is  10  per 
cent.  ?  What,  when  it  is  8^-  per  cent.  ? 

235-240.  Reduce  the  following  sums,  Sterling  Money,  at 
custom-house  valuation,  to  U.  S.  currency  :  £9  5s.  ;  £27  3s. 
4d.  3qr.  ;  £39  Qd. ;  £270  14s.  Qd.  2qr.  ;  £4180  12s.  -Sd. ; 
£69480  9c?. 

241-258.  Find  the  value  of  each  of  the  preceding  sums, 
in  U.  S.  currency,  when  the  premium  of  exchange  on  Eng- 
land is  8  per  cent.  ;  is  9  per  cent.  ;  is  10  per  cent. 

259-263.  Reduce  $100'20  to  Canada  and  to  the  different 
State  currencies. 

264-268.  Reduce  $3 7 '3 7  to  Canada  and  State  curren- 
cies. 

269-273.  Reduce  $1000  to  its  equivalent  value  in  Can- 
ada and  State  currencies. 

274-278.  Reduce  £75  15s.  6d.  of  the  respective  curren- 
cies mentioned  in  the  table  to  Federal  Money. 


§  122.]        REDUCTION  OF  CURRENCIES.  213 

279-283.  Reduce  £80  55.  3c?.  of  the  different  currencies 
to  Federal  Money. 

284-288.  Reduce  £1000  of  the  different  currencies  to 
Federal  Money. 

289.  How  many  sovereigns  in  $8496  ? 

290.  How  many  5 -franc  pieces  in  $10765? 

291-295.  In  $9284'47  how  many  Mexican  doubloons? 
how  many  10-Thaler  pieces  ?  how  many  Canada  pounds  ? 
how  many  rupees  of  Bengal  ?  how  many  ducats  of  Naples  ? 

296-301.  Reduce  to  Federal  Money  7498  rix-dollars  of 
Bremen  ;  25480  rupees  of  Bengal ;  4879^  silver  roubles  of 
Russia;  79682  sovereigns;  729810f  pagodas  of  India; 
1987629  francs. 

302.  Suppose  I  owe  a  Liverpool  merchant  £17496  85., 
what  sum  in  Federal  Money  must  I  pay  him,  when  exchange 
on  England  is  9  per  cent,  premium  ? 

303.  I  am  indebted  to  a  Liverpool  house  in  the  sum  of 
$25000*75.     How  many  pounds  sterling  must  I  pay  to  his 
order,  when  exchange  on  England  is  1 0  per  cent,  premium  ? 

304.  A  New  England  merchant  wished  to  pay  £784  105., 
Georgia  currency,  to  a  merchant  in  Savannah.     What  sum 
in  N.  E.  currency  must  he  remit? 

305.  How  many  5 -franc  pieces  must  a  Paris  house  remit 
to  pay  £9841  7s.,  N.  Y.  currency? 

306.  The  rate  of  duty  on  imported  dried  plums,  in  1842, 
was  £1  8s.  per  cwt.     How  much  is  that  per  lb.,  U  S.  cur- 
rency ? 

307.  The  duty  on  grain,  not  rated  as  corn  or  seeds,  was 
185.  per  cwt.     What  is  that  per  cwt.,  U.  S.  currency  ? 

308.  The  duty  on  rose-wood  was  £6  per  ton.     Wliat  is 
that  per  cwt.,  U.  S.  currency  ? 

309-314.  In  $1000  how  many  Ounces  of  Sicily  ?  how 
many  ducats  of  Naples  ?  how  many  florins  of  Augsburg  ? 


214:  PROPORTION.  [CHAP.  XIV. 

how  many  rix-dollars  of  Bremen  ?  how  many  Mexican  doub- 
loons ?  how  many  Louis  d'ors  ? 

315-325.  In  1000  Mexican  doubloons  how  many  dollars  ? 
how  many  crowns  ?  how  many  sovereigns  ?  how  many  spe- 
cie-dollars of  Denmark  ?  how  many  specie-dollars  of  Nor- 
way ?  how  many  pagodas  of  India  ?  how  many  rupees  of 
Bengal  ?  how  many  milrees  of  Portugal  ?  how  many  Mark 
bancos  of  Hamburg  ?  how  many  English  guineas  ?  how 
many  francs  ? 


CHAPTER    XIV. 

PROPORTION. 

§  123.  WHEN  the  ratio  of  two  quantities  is  the  same  as 
the  ratio  of  two  other  quantities,  the  four  quantities  are  in 
proportion.  Thus,  the  ratio  of  8  yards  to  4  yards  is  the 
same  as  the  ratio  of  1 2  dollars  to  6  dollars  ;  therefore,  there 
is  a  proportion  between  8  yards,  4  yards,  12  dollars,  and  6 
dollars. 

The  usual  method  of  denoting  that  four  terms  are  in  pro- 
portion, is  by  means  of  points  or  dots.  Thus,  the  above 
proportion  is  written, 

8  yards  :  4  yards  :  :  12  dollars  :  6  dollars. 

Where  two  dots  are  placed  between  the  first  and  second 
terms,  and  between  the  third  and  fourth  ;  and  four  dots  are 
placed  between  the  second  and  third. 

The  two  dots  are  equivalent  to  the  sign  of  division,  and 
the  four  dots  correspond  with  the  sign  of  equality.  Thus, 
the  above  proportion  may  be  written, 

8  yards H- 4  yards  =  12  dollars -f- 6  dollars. 


§  123.]  PROPORTION.  215 

Either  of  the  foregoing  forms  of  this  proportion  may  be 
read, 

8  yards  is  to  4  yards  as  12  dollars  is  to  6  dollars. 

The  first  term  of  a  ratio  is  called  the  antecedent ;  the  sec- 
ond is  called  the  consequent. 

The  first  and  fourth  terms  of  a  proportion  are  called  the 
extremes  ;  the  second  and  third  terms  are  called  the  means. 

Since  in  a  proportion  the  quotient  of  the  first  term  di- 
vided by  the  second,  is  equal  to  the  quotient  of  the  third 
term  divided  by  the  fourth,  we  have,  using  the  above  pro- 
portion, l-nr-1^-.  If  we  reduce  the  fractions  to  a  common 

8X6     12x4 
denominator,  they  become = ,    or   omitting   the 

common  denominator  4x6,  which  is,  in  effect,  multiplying 
each  fraction  by  4  X  6,  we  have  8  X  6,  or  48  =  1 2  X  4  or  48  ; 
that  is,  the  product  of  the  extremes  is  equal  to  the  product  of 
the  means. 

8X6=48              _  8X6  =  48 
Again,  — =4,  and ^ =  12. 

Hence,  if  the  product  of  the  extremes  be  divided  by  either 
mean,  the  quotient  will  be  the  other  mean.* 

12X4  j!2x4 

Again,  —^—  =  6,  and      Q     =8. 

Hence,  if  the  product  of  the  means  be  divided  by  either 
extreme,  the  quotient  will  be  the  other  extreme. 

*  It  is  often  required  to  find  a  mean  proportional  when  the  extremes  are  given ; 
that  is,  one  mean  of  a  proportion  iij  which  the  means  are  equal.  Thus,  4  and  9 
being  the  extremes,  give  a  product  of  36,  which  is  equal  to  the  product  of  the  means. 
Hence  the  means  may  be  2  and  18,  3  and  12,  or  6  and  6  ;  of  these  6  and  6  are  the 
equal  means  ;  thus,  4  :  G  :  :  6  :  9. 

Therefore,  to  find  a  mean  proportional  when  the  extremes  are  given,  the  square 
root  of  the-'r  product  must  be  found ;  that  is,  the  number  which  being  multiplied 
by  itself  will  produce  that  product. 


216  SIMPLE  PROPORTION.  [CHAP.  XIV 

From  the  above  properties,  we  see  that  if  any  three  of  th 
four  terms  which  constitute  a  proportion  are  given,  the  re- 
maining term  can  be  found. 

The  method  of  finding  the  fourth  term  of  a  proportion, 
when  three  terms  are  given,  constitutes  the  RULE  OF  THREE. 

Let  us  now  apply  what  has  been  explained. 

If  8  yards  of  cloth  are  worth  $12,  what  are  24  yards 
worth  ? 

The  value  sought  must  be  as  many  times  greater  than 
$12,  as  24  yards  is  greater  than  8  yards.  Hence,  there  is 
the  same  ratio  between  $1A2  and  the  value  sought,  as  there- 
is  between  8  yards  and  24  yards.  Consequently,  we  have 
this  proportion : 

8  yards  :   24  yards  :  :  $12  :  value  sought. 

Taking  the  product  of  the  means,  we  have  24  X  12  =  288. 
This,  divided  by  the  first  term,  which  is  one  of  the  extremes, 
gives  -2.jp-=36  for  the  other  extreme  or  fourth  term  sought, 
which  must  be  of  the  same  kind  as  the  third  term  ;  there- 
fore $36  is  the  value  of  24  yards. 

NOTE. — When  we  take  the  product  of  the  means  we  do  not  multi- 
ply the  24  yards  by  12  dollars,  hut  simply  multiply  24,  the  number 
denoting  the  yards,  by  12,  the  number  denoting  the  dollars.  The 
product,  288,  is  neither  yards  nor  dollars,  but  288  units.  When  we 
divide  this  product  by  the  first  term  of  the  proportion,  we  do  not 
divide  by  8  yards,  but  simply  by  8,  the  number  denoting  the  yards. 
The  quotient,  36,  gives  the  fourth  term  of  the  proportion  ;  and  since 
the  fourth  term  is  of  the  same  denominate  value  as  the  third  term, 
our  fourth  term,  or  answer,  must  be  36  dollars. 

From  the  foregoing  explanations,  we  deduce  this  first 
form  of  the 

RULE  FOR  SIMPLE  PROPORTION,  OR  SINGLE  RULE  OF  THREE. 

I.  Form  a  proportion  by  placing  for  the  third  term  the 

quantity  which  is  of  the  same  denomination  as  the  answer 


J  123.]  SIMPLE  PROPORTION.  217 

sought.  Of  the  two  remaining  quantities,  the  larger  must 
be  taken  for  the  second  term,  when  the  answer  is  to  exceed 
the  third  term  ;  but  the  smaller  must  be  taken  for  the  second 
term,  when  the  answer  is  to  be  less  than  the  third  term. 

II.  Having  written  the  three  terms  of  the  proportion,  or, 
as  usually  expressed,  having  stated  the  question,  then  multi- 
ply the  second  and  third  terms  together,  and  divide  the  prod- 
uct by  the  first  term. 

NOTE. — Since  there  is  a  ratio  between  the  first  and  second  terms, 
they  must  be  reduced  to  the  same  denominate  value.  Also,  the 
third  term  must  be  reduced  to  its  lowest  denomination ;  then  the 
quotient  found  by  dividing  the  product  of  the  means  by  the  first- 
term  will  be  of  the  same  denomination  as  the  third  term. 

EXAMPLES. 

1.  If  25  Ibs.  of  coffee  cost  $3  '2 5,  what  will  312  Ibs.  cost  ? 

2.  What  cost  6  cords  of  wood,  at  $7  for  2  cords  ? 

3.  What  will  9  pairs  of  shoes  cost,  if  5  pairs  cost  £2 
25.  Qd.  ? 

4.  If  there  are  9  weeks  in  63  days,  how  many  weeks  in 
365  days  ? 

5.  If  a  railroad  cargoes  17  miles  in  45  minutes,  how  far 
will  it  go  in  5  hours  ?       , 

6.  If  $100  will  gain  $7  in  one  year,  how  long  will  it  re- 
quire to  gain  $100  ? 

7.  If  3  paces  or  common  steps  of  a  person  are  equal  to 
2  yards,  how  many  yards  will  480  paces  make  ? 

8.  If  15  men  can  raise  a  wall  of  masonry,  12  feet,  in  one 
week,  how  many  will  be  necessary  to  raise  it  20  feet  in  the 
same  time  ? 

9.  If  7  tons  of  coal,  of  2000  pounds  each,  will  last  3^ 
months,  of  30  days  each,  how  much  will  be  consumed  in  3 
weeks,  or  21  days? 

19 


218  PROPORTION.  [CHAP,  xiv 

10.  If  9J  bushels  of  wheat  make  2  barrels  of  flour,  how 
many  bushels  will  be  required  to  make  1  3  barrels  ? 

11.  If  a  steamboat  of  242  feet  in  length  move  15  miles 
in  one  hour,  how  many  seconds  will  it  require  to  move  its 
own  length  ? 

12.  If  a  steamboat  of  242  feet  in  length  move  15  miles 
an  hour,  how  many  times  its  own  length  will  it  move  in  1  1 
hours  ? 

13.  A  reservoir  has  a  pipe  capable  of  discharging  30  gal- 
lons in  one  minute.     What  time  will  be  necessary  to  dis- 
charge 1  5  hogsheads  ? 

14.  If  a  man  can  mow  9  acres  of  grass  in  3  J  days,  of  10 
hours  each,  how  long  will  it  require  for  him  to  mow  21  acres  ? 

15.  If  100  pounds  of  galena,  or  lead  ore,  yield  83  pounds 
of  pure  metal,  how  much  pure  metal  will  7  tons  of  galena 
produce,  if  we  reckon  2240  pounds  to  the  ton  ? 

If  12  barrels  of  flour  are  worth  $54,  what  is  the  value  of 
42  barrels  at  the  same  rate  ? 

In  this  example  it  is  obvious  that  2  times  12  barrels  would  be 
worth  2  times  $54  ;  3  times  12  barrels  would  be  worth  3  times  $54  ; 
4  times  12  barrels  would  be  worth  4  times  $54,  and  so  on  for  other 
ratios.  *  The  ratio  of  42  barrels  to  12  barrels  is  i|. 

If  we  multiply  $54  by  this  ratio,  it  will  evidently  give  the  value 
of  42  barrels. 


"We  may  now  employ  the  same  rules  for  simplifying  this  expres- 
sion as  were  used  under  §  114;  that  is  to  say,  vre  may  reject  such 
factors  as  are  common  to  both  numerators  and  denominators.  Thus, 
dividing  the  denominator  12,  and  the  numerator  42,  each  by  6,  it 
becomes 

7 
jfi 

XT-,  or$54x$. 

A.  ft 


§  123.]  SIMPLE  PROPORTION.  219 

Again,  dividing  the  denominator  2  and  $54  of  numerator  each  by 
2,  we  have 

27 

$£xl,  or  $27x7  =  $189.  Ans. 


If  200  sheep  yield  650  pounds  of  wool,  how  many  pounds  will  825 
sheep  yield  ? 

In  this  example,  the  answer  is  required  to  be  in  pounds  ;  we  there- 
fore take  650  pounds  for  the  third  term.  The  ratio  of  825  sheep  to 
200  sheep  is  |||.  Hence  we  have 

650Z6.  X 
Cancelling,  we  have 

33 


-,  or,  650/6. 

8 
Again,  cancelling,  we  have 


4 

If  11  of  a  pound  of  sugar  cost  %%  of  a  shilling,  how  much  will  •£§ 
of  a  pound  cost  ? 

In  this  example,  our  third  term  is  f  £  of  a  shilling.  And  since  -^ 
of  a  pound  is  less  than  11,  we  must  obtain  our  ratio  by  dividing  ^ 
by  11,  which  gives  ^  X  f  f .  Multiplying  the  third  term  by  this  ratio, 
we  have  ff  of  a  shilling  X—  Xjf  To  reduce  this  with  the  least 
labor,  we  must  resort  to  the  method  of  cancelling.  Thus,  cancelling 
the  23,  which  occurs  in  both  numerator  and  denominator,  also  13  of 
the  numerator  against  a  part  of  the  26  of  the  denominator,  our  ex- 
pression will,  by  this  means,  become  -J  of  a  shilling  XfXyly=^ 
of  a  shilling. 

NOTE. — This  method  of  cancelling  should  be  used  when  the  nature 
of  the  question  will  admit,  since  it  will  always  simplify  the  operation. 

From  the  above  explanation,  we  deduce  this  second  form 
of  the 


220  PROPORTION.  [CHAP.  xrv. 

RULE    OF    THREE. 

Of  the  three  quantities  which  are  given,  one  will  always 
be  of  the  same  kind  as  the  answer  sought ;  this  quantity  will 
be  the  third  term.  Then,  if  by  the  nature  of  the  question,  the 
answer  is  required  to  be  greater  than  the  third  term,  divide 
the  greater  of  the  two  remaining  quantities  by  the  less,  for 
a  ratio  ;  but  if  the  answer  is  required  to  be  less  than  the  third 
term,  then  divide  the  less  of  the  two  remaining  quantities  by 
the  greater,  for  a  ratio.  Having  obtained  the  ratio,  multi- 
ply the  third  term  by  it,  and  it  will  give  the  answer  in  the 
same  denomination  as  is  the  third  term. 

NOTE. — Before  obtaining  the  ratio,  by  means  of  the  first  two  terms, 
we  must  reduce  them  to  like  denominations.     See  §  116. 

EXAMPLES. 

16.  If  a  tree  38  feet  9  inches  in  height,  give  a  shadow  of 
49  feet  2  inches,  how  high  is  that  tree  which,  at  the  same 
time,  casts  a  shadow  of  71  feet  7  inches? 

17.  If  3^-  pounds  of  coffee  cost  2j  shillings,  how  much 
will  IQi  pounds  cost  ? 

18.  If  6  men  earn  $25  in  6  days,  how  much  can  they 
earn  in  25  days  ? 

19.  If  a  locomotive  move  95  miles  in  4  hours,  how  far 
does  it  go  each  hour  ? 

20.  If  it  take  10  hours  for  6  men  to  do  a  piece  of  work, 
how  long  will  it  take  15  men  to  do  the  same  work  ? 

21.  Gave  $72  for  11  barrels  of  fish.     How  much  will  88 
barrels  cost  at  the  same  rate  ? 

22.  If  431  pounds  of  cheese  cost  $2'20,  what  will  216-§ 
pounds  cost  at  the  same  rate  ? 

23.  If  I  pay  $3 -90  for  sawing  7  cords  of  wood,  how  much 
ought  I  to  give  for  sawing  23-J-  cords  ? 


§  123.]  SIMPLE  PROPORTION.  221 

24.  If  T%  of  a  ship  is  worth  $2853,  what  is  the  whole 
worth  ? 

25.  If  T4g  of  my  income  is  $533,  what  is  my  whole  income  ? 
20.  A  person  failing  in  business,  finds  that  he  owes  $7560, 

and  that  he  only  has  $3100  to  pay  the  debt  with.     How 
much  can  he  pay  to  that  creditor  whose  claim  is  $756  ? 

27.  If  it  require  5^  bushels  of  wheat  to  make  one  barrel 
of  flour,  how  many  bushels  will  be  required  for  100  barrels 
of  flour? 

28.  If  7  barrels  of  flour  are  sufficient  for  a  family  6  months, 
how  many  barrels  will  they  require  for  1 1  months  ? 

29.  If  it  take  25  yards  of  carpeting,  a  yard  wide,  to  cover 
a  certain  floor,  how  many  yards  of  -|  carpeting  will  be  ne- 
cessary to  cover  the  same  floor  ? 

30.  If  a  person  travel  8  miles  in  10  hours,  how  far  will 
he  travel  in  5  days,  by  travelling  8  hours  each  day  ? 

31.  If  35   pounds  of  feathers  cost  $15,  what  will  100 
pounds  cost  at  the  same  rate  ? 

32.  If  a  man  perform  a  certain  piece  of  work  in  18  days, 
when  he  works  8  hours  per  day,  how  many  days  will  he 
require  if  he  work  1 0  hours  each  day  ? 

33.  If  a  piece  of  board  12  inches  wide  and  12  inches 
long  make  one  square  foot,  how  many  inches  of  length  must 
be  taken  from  a  board  15  inches  wide  to  make  a  square  foot  ? 

34.  If  8  men  can  mow  a  field  in  5  days,  in  how  many 
days  can  5  men  mow  it  ? 

35.  If  27  J  yards  of  cloth  cost  $60,  how  many  yards  can 
I  buy  for  $100? 

36.  If  271  yards  of  cloth  cost  $60,  what  will  45f  yards 
cost? 

37.  'If  fof  a  ship  is  worth  $9000,  what  is  her  whole  value '? 

38.  If  T3F  of  a  city  lot  is  sold  for  $500,  what  would  T7^ 
of  the  same  lot  sell  for  at  the  same  rate  ? 

19* 


222  PROPORTION.  [CHAP  xiv 

39.  Admitting  that  the  earth  moves  in  its  orbit  about  the 
sun,  a  distance  of  597000000  miles,  in  365  days  6  hours, 
how  far  on  an  average  does  it  move  each  hour  ? 

40.  If  the  diurnal  rotation  of  the  earth  move  its  equato- 
rial portions  about  24900  miles  each  day,  how  far  is  that  in 
each  hour  ? 

41.  If  it  require  10  years  of  365J  days,  for  light  to  pass 
frcto  a  fixed  star  to  the  earth,  how  many  miles  distant  is  it 
on  the  supposition  that  light  moves  192090  miles  in  one 
second  ? 

42.  If  by  a  leak  of  a  ship,  f  enough  water  run  in,  in  4 
hours,  to  sink  her,  how  long  can  she  survive  ? 

43.  If  I  pay  $25  for  the  masonry  of  4000  bricks,  how 
much  ought  I  to  pay  for  the  work  which  requires  100000 
bricks  ? 

44.  If  a  steamship  require  14  days  to  sail  a  distance  of 
3000  miles,  what  time,  at  the  same  rate  of  sailing,  would 
she  require  to  sail  24900  miles  ? 

45.  Admitting  the  diameter  of  the  earth  to  be  8000  miles 
and  the  loftiest  mountain  to  be  5  miles  in  height,  what  ele- 
vation must  be  made  on  a  globe  of  16  inches  diameter  to 
represent  accurately  the  height  of  such  mountain  ? 

46.  If  $100  in  12  months  bring  an  interest  of  $7,  how 
much  will  be  the  interest  of  $100  for  8  months? 

47.  If  the  interest  of  $100  for  12  months  is  $7,  what  will 
be  the  interest  of  $75  for  the  same  time  ? 

48.  If  in  12  months  the  interest  of  $100  is  $7,  how  long 
must  $100  be  on  interest  to  gain  $10  ? 

49.  If  a  glacier  of  60  miles  in  length  move  50  inches  per 
annum,  in  what  time  will  it  move  its  whole  length  ? 

50.  If  a  staff  of  10  feet  in  length  give  a  shadow  of  15  feet, 
how  high  is  that  tree  whose  shadow  measures  90  feet  ? 

51.  Suppose  sound  to  move  1100  feet  in  a  second,  how 


§  123.]  SIMPLE  PROPORTION. 

many  miles  distant  is  a  cloud,  in  which  lightning  is  observed 
16  seconds  before  the  thunder  is  heard,  no  allowance  being 
made  for  the  motion  of  light  ? 

52.  If  it  require  30  yards  of  carpeting  which  is  ^  of  a 
yard  wide  to  cover  a  floor,  how  many  yards  of  carpeting 
which   is   lj  yards  wide  will  be  necessary  to    cover  the 
same  floor  ? 

53.  If  the  earth  move  through  12  signs,  or  360°,  in  365-J- 
days,  how  far  will  it  move  in  a  lunar  month  of  29J  days  ? 

54.  Suppose  a  steamboat    capable  of   making  15  miles 
each  hour,  to  move  with  a  current  whose  velocity  is  2^  miles 
per  hour,  what  will  be  the  whole  distance  made  during  13^ 
hours  ?     And  what  distance  will  the  boat  move  in  the  same 
time  against  the  same  current  ? 

55.  If   the  magnetic   influence   move  through  the  tele- 
graphic wires  at  the  rate  of  200000  miles  in  one  second  of 
time,  how  many  times  could  it  pass  around  the  world  in 
one  second,  allowing  the  circumference  of  the  earth  to  be 
24899  miles? 

56.  If  A.  can  do  a  piece  of  work  in  7  days,  and  B.  can 
do  it  in  8  days,  what  part  of  it  can  both  do  in  3  J  days  ? 

57.  A  reservoir,  whose  capacity  is  1000  hogsheads,  has 
a  supply-pipe,  by  means  of  which  it  receives  3-00  gallons 
each  hour ;  it  also  has  two  discharging  pipes,  the  first  of 
which  discharges  |-  of  a  gallon  each  minute,  the  second  dis- 
charges 1}  gallons  per  minute.     The  reservoir  being  empty, 
in  what  time  will  it  be  filled  if  the  supply-pipe  alone  is 
opened  ?     In  what  time,  if  the  supply-pipe   and  the  first 
discharging  pipe  are  opened  ?     In  what  time,  if  the  supply- 
pipe  and  the  second  discharging  pipe  ?     And  in  what  time 
if  all  three  are  opened  ? 


224:  PROPORTION.  [CHAP.  XIV. 

COMPOUND  PROPORTION. 

§  124.  A  Compound  Proportion  is  an  expression  of  the 
equality  between  the  product  of  several  ratios  and  a  simple 
ratio. 

If  6  men  can  mow  30  acres  of  grass  in  5  days,  by 
working  8  hours  each  day,  how  many  acres  can  4  men  mow 
in  9  days  of  10  hours  each  ? 

Before  performing  this  example  by  the  Rule  of  Compound 
Proportion,  or  Double  Rule  of  Three,  let  us  solve  it,  first 
by  Analysis  and  then  by  Ratio. 

ANALYSIS. — If  6  men  can  mow  30  acres  in  5  days  of  8 
hours  each,  1  man  will  mow  -^p~=5  acres  in  5  days  of  the 
same  length  :  4  men  will  mow  4x5  acres  or  20  acres,  in 
5  days. 

If  4  men  mow  20  acres  in  5  days,  in  1  day  they  will 
mow  -2-£r=±  acres  ;  in  9  days,  4  X  9  =  36  acres. 

If  4  men  mow  36  acres  in  9  days  of  8  hours  each,  they 
will  mow  -^-  or  4j  acres,  by  working  but  1  hour  each  day, 
and  4jxlO  or  45  acres,  by  working  10  hours  each  day. 
The  answer,  then,  is  45  acres. 

NOTE. — In  this  Complex  Analysis,  as  in  Simple  Analysis,  we  reason 
from  the  given  quantity  back  to  1,  then  from  1  to  the  quantity  re- 
quired. 

RATIO. — Had  the  number  of  days,  as  well  as  hours  in 
each  day  been  the  same  in  both  cases,  the  question  would 
have  been  equivalent  to  the  following  : 

If  6  men  mow  30  acres  of  grass,  how  many  acres  will 
4  men  mow  ? 

It  is  evident  the  number  of  acres  sought  would  be  the 
same  fractional  part  of  30  acres  that  4  men  is  of  6  men ; 
that  is,  the  quantity  required  is 

of  30  acres. 


§  124.]         COMPOUND  PROPORTION. 

If,  now,  we  take  into  account  the  number  of  days,  still 
supposing  the  number  of  hours  in  each  day  to  remain  the 
same  in  both  cases,  our  question  would  become  : 

If  -g-  of  30  acres  can  be  mowed  in  5  days,  how  much  can 
be  mowed  in  9  days  ? 

The  answer  in  this  case  is  obviously 

f  of  £  of  30  acres. 

Now,  taking  into  account  the  number  of  hours  in  each 
day,  our  question  will  become  as  follows  : 

If  §  of  |  of  30  acres  can  be  mowed  in  a  certain  time, 
when  8  hours  are  reckoned  to  each  day,  how  much  could 
be  mowed  when  10  hours  are  reckoned  to  each  day? 

This  leads  to  the  following  final  result  : 

J^_  of  f  of  |  of  30  acres. 
By  cancelling,  we  reduce  this  last  expression  to  45  acres. 

As  a  second  example,  we  will  solve  the  following  by  ratio  : 

500  men,  working  12  hours  each  day,  have  been  employed 
57  days  to  dig  a  canal  of  1800  yards  long,  7  yards  wide, 
and  3  yards  deep,  how  many  days  must  860  men,  working 
10  hours  each  day,  be  employed  in  digging  another  canal 
of  2900  yards  long,  12  yards  wide,  and  5  yards  deep,  in  a 
soil  which  is  3  times  as  difficult  to  excavate  as  the  first  ? 

In  this  example,  the  odd  term  is  57  days. 

The  different  ratios  will  be  as  follows  : 
Of  t 


—   f  ratio  of  the  hours. 


—  2|  ratio  of  lengths  of  the  canals. 

-J-—  ratio  of  widths  of  the  canals. 

^=  ratio  of  depths  of  the  canals. 

J=  ratio  of  the  difficulty  in  excavation. 


22j5  PROPORTION.  [CHAP.  xrv. 

Multiplying  successively  these  ratios  and  the  odd  term, 
we  have 

57  daysxffxf  XffX-y-Xfxf. 

This  becomes,  after  cancelling  factors, 


From  the  above  work  we  see  that  questions  of  Compound 
Proportion  may  be  solved  by  the  following 

RULE. 

Among  the  terms  given,  there  will  be  but  one  like  the  an- 
swer, which  we  will  call  the  odd  term.  The  other  terms  will 
appear  in  pairs  or  couplets.  Form  ratios  out  of  each  couplet 
in  the  same  manner  as  in  the  Rule  of  Three  ;  then  multiply 
all  the  ratios  and  the  odd  term  together,  and  it  will  give  the 
answer  in  the  same  name  and  denomination  as  the  odd  term. 

NOTE.  —  Before  forming  ratios  from  the  couplets,  they  must  be  re- 
duced to  the  same  denominate  value.  See  §  115. 

EXAMPLES. 

Solve  the  following  problems,  first  by  Analysis,  and  then 
by  the  Rule  for  Compound  Proportion  : 

58.  If  a  person  travel  300  miles  in  17  days,  journeying 
6  hours  each  day,  how  many  miles  will  he  travel  in  15  days, 
journeying  10  hours  a  day  ? 

59.  If  a  marble  slab  10  feet  long,  3  feet  wide,  and  3 
inches  thick,  weigh  400  pounds,  what  will  be  the  weight 
of  another  slab  of  the  same  marble,  whose  length  is  8  feet, 
width  4  feet,  and  thickness  5  inches  ? 

60.  If  the  expenses  of  a  family  of  10  persons  amount  to 
$250  in  23  weeks,  how  long  will  $600  support  a  family  of 
8  persons  at  the  same  rate  ? 

61.  15  men,  working  10  hours  each  day,  have  employed 


§  124.]         COMPOUND  PROPORTION.  227 

18  days  to  build  450  yards  of*stone  fence.  How  many 
men,  working  12  hours  each  day,  for  8  days,  will  be  re- 
quired to  build  480  yards  of  similar  fence  ? 

62.  If  it  require  1200  yards  of  cloth  -|  wide  to  clothe 
500  men,  how  many  yards  which  is  |-  wide  will  it  take  to 
clothe  960  men  ? 

63.  If  8  men  will  mow  36  acres  of  grass  in  9  days,  by 
working  9  hours  each  day,  how  many  men  will  be  required 
to  mow  48  acres  in  12  days,  by  working  12  hours  each  day  ? 

64.  If  11  men  can  cut  49  cords  of  wood  in  7  days,  when 
they  work  14  hours  per  day,  how  many  men  will  it  take  to 
cut  140  cords  in  28  days,  by  working  10  hours  each  day? 

65.  If  12  ounces  of  wool  make  2j-  yards  of  cloth  that  is 
6  quarters  wide,  how  many  pounds  of  wool  will  it  take  for 
150  yards  of  cloth,  4  quarters  wide? 

66.  If  the  wages  of  6  men  for  14  days  be  84  dollars, 
what  will  be  the  wages  of  9  men  for  16  days  ? 

67.  If  100  Inen  in  40  days  of  10  hours  each,  build  a  wall 
30  feet  long,  8  feet  high,  and  2  feet  thick,  how  many  men 
must  be  employed  to  build  a  wall  40  feet  in  length,  6  feet 
high,  and  4  feet  thick,  in  20  days,  by  working  8  hours  each 
day? 

68.  In  how  many  days,  working  9  hours  a  day,  will  24 
men  dig  a  trench  420  yards  long,  5  yards  wide,  and  3  yards 
deep,  if  248  men,  working  11  hours  a  day,  in  5  days  dig  a 
trench  230  yards  long,  3  yards  wide,  and  2  yards  deep  ? 

69.  Suppose  that  50  men,  by  working  5  hours  each  day, 
can  dig,  in  54  days,  24  cellars,  which  are  each  36  feet  long, 
21  feet  wide,  and  10  feet  deep,  how  many  men  would  be 
required  to  dig,  in  27  days,  18  cellars,  which  are  each  48 
feet  long,  28  feet  wide,  and  9  feet  deep,  provided  they  work 
only  3  hours  each  day  ? 

70.  If  40  yards  of  cloth  3  quarters  wide  cost  $45,  what 

14 


228  PROPORTION.  [CHAP.  xiv. 

will  36  yards  of  the  same  quality  cost,  which  is  5  quarters 
wide? 

71.  If  8400  will  in  9  months  gain  $21,  when  the  rate  of 
interest  is  7  per  cent,  per  annum,  how  much  will  $360  gain 
in  8  months,  if  the  rate  per  cent,  is  6  ? 

72.  If   $400  require  9  months  to    gain  $21,  when  the 
rate  per  cent,  is  7,  how  long  a  time  will  $360  require  to 
gain  $14-40,  if  the  rate  per  cent,  is  6  ? 

73.  If  $400,  to  gain  $21  in  9  months,  require  a  rate  of 
7  per  cent.,  what  must  be  the  rate  per  cent,  for  $360  to 
gain  $14-40  in  8  months  ? 

74.  If  it  require  $400  to  gain  $21  in  9  months,  when 
the  rate  per  cent,  is  7,  how  much  will  be  required  to  gain 
$14-40  in  8  months,  when  the  rate  per  cent,  is  6  ? 

75.  If  the  freight  on  72  barrels  of  flour,  for  a  distance 
of  95  miles,  is  $7 -60,  what  would  it  be  at  the  same  rate 
on  120  barrels  for  a  distance  of  144  miles  ? 

76.  If  from  a  dairy  of  30  cows,  each  furnishing  16  qts. 
of  milk  daily,  24  cheeses  of  55  pounds  each  are  made,  in 
36  days,  how  many  cows  will  be  required,  if  each  gives  4^ 
gallons  of  milk  daily,  to  produce  in  30  days  33  cheeses  of 
100  pounds  each  ? 

77.  If  I  pay  $45  for  40  yards  of  cloth  which  is  3  quar- 
ters wide,  how  many  yards  of  the  same  quality  of  cloth 
which  is  5  quarters  wide  ought  $60  to  buy  ? 

78.  If  6  persons  eat  21  dollars'  worth  of  bread  in  4 
months,  when  flour  is  sold  at  $7  per  barrel,  in  how  many 
months  will  10  persons  eat  50  dollars'  worth,  when  flour  is 
$5  per  barrel  ? 

79.  If  21  dollars'  worth  of  bread,  when  flour  is  $7  per 
barrel,  will  supply  6  persons  4  months,  how  many  persons 
can  be  supplied  for  8  months  for  50  dollars,  when  flour  is 
$5  per  barrel  ? 


§  125.]  ARBITRATION  OF  EXCHANGE.  229 

80.  If  21  dollars'  worth  of  bread,  when  flour  is  $7  per 
barrel,  will  supply  6  persons  for  4  months,  how  many  dol- 
lars' worth  will  be  required  to  supply  10  persons  during  8 
months,  if  flour  is  85  per  barrel  ? 

ARBITRATION  OF  EXCHANGE. 

§  125.  It  sometimes  happens  that  a  merchant  desires  to 
pay  a  debt  to  a  foreign  creditor  through  three  or  four  agents 
or  brokers  in  different  countries.  There  must,  m  such  cases, 
be  a  chain  of  exchanges  called  arbitration  of  exchange  ;  for 
example : 

A  merchant  in  Denmark  owes  a  New  York  merchant 
$280.  How  many  specie- dollars  of  his  own  country  must 
he  remit  through  houses  in  Hamburgh,  Amsterdam,  and  St. 
Petersburgh,  if  $0'35  =  1  mark  banco;  8  marks =  7  guil- 
ders ;  15  guilders =8  silver  roubles  ;  21  roubles  =  15  specie- 
dollars  of  Denmark  ? 

28000  X  3*5  X  |-  X  T85  X  2T>  which,  after  cancelling,  be- 
comes 266J-,  the  number  of  specie-dollars.  Since  35  cts.= 
1  mark,  there  will  be  ^  as  many  marks  as  cents  ;  as  8 
marks  =  7  guilders,  there  will  be  |-  as  many  guilders  as 
marks  ;  in  like  manner  there  will  be  -fj  as  many  roubles  as 
guilders,  and  if  as  many  specie-dollars  as  roubles. 

It  will  be  seen  that  examples  of  this  kind  require  the 
multiplication  of  a  given  term  by  the  product  of  a  series 
of  ratios  ;  and  that  the  process  is  the  same  as  that  of  Com- 
pound Proportion,  §  124.  No  independent  rule  is  necessary. 

EXAMPLES. 

81-83.  A  New  York  merchant  owes  in  London  £375. 
How  many  dollars  must  he  remit  through  houses  in  Naples 
and  in  Paris,  if  £20=121  ducats  ;  90  ducats =41 4  francs  ; 

20 


230  PROPORTION.  [CHAP.  xrv. 

500  francs=91  dollars  ?  How  many  dollars  must  he  remit 
directly  to  London  if  the  premium  of  exchange  is  9  per 
cent,  above  the  par  value  of  $4£  to  the  £  ?  And  how 
much  was  lost  by  the  former  circuitous  method  of  remit- 
ting ? 

84-86.  A  merchant  in  New  York  orders  £1000  due  him 
in  London  to  be  forwarded  to  him  by  the  following  route : 
to  Hamburgh,  at  15  mark  bancos  per  £ ;  thence  to  Copen- 
hagen, at  100  mark  bancos  for  43  rix-dollars;  thence  to 
Bourdeaux,  at  4  rix-dollars  for  18  francs;  thence  to  New 
York,  at  500  francs  for  93  dollars.  How  many  dollars  did 
he  receive  ?  How  many  dollars  would  he  have  received 
had  he  ordered  the  £1000  direct  to  him,  the  premium  of 
exchange  being  7  per  cent,  above  the  par  value  of  $4|-  to 
the  £  ?  And  how  much  did  he  save  by  the  above  circui- 
tous route  ? 

87.  If  a  man  receives  $27  for  16  barrels  of  cider,  and  he 
can  buy  2  barrels  of  flour  for  $11,  and  3  tons  of  coal  .for 
4  barrels  of  flour,  and  45  pounds  of  tea  for  2  tons  of  coal, 
how  many  pounds  of  tea  ought  he  to  receive  for  7  barrels 
of  cider  ? 

88.  If  25  pears  can  be  bought  for  10  lemons,  and  28  lem- 
ons for  18  pomegranates,  and  1  pomegranate  for  48  almonds, 
and  50  almonds  for  70  chestnuts,  and  108  chestnuts  for  2j 
cents,  how  many  pears  can  I  buy  for  $1*35  ? 

89.  If  121  English  guineas  are  equal  to  125  pounds  ster- 
ling, and  £23  =  61  pagodas  of  India,  and  5  pagodas =$9, 
how  many  English  guineas  will  be  equal  to  $100  ? 

90.  If  74  francs  =  9  tales  of  China,  10  tales  of  China=6 
ounces  of  Sicily,  and  5  ounces  of  Sicily =$12,  how  many 
francs  will  be  equal  to  $172  ? 


§  126.]  PARTNERSHIP.  231 


PARTNERSHIP  OR  FELLOWSHIP. 

§  126.  Fellowship  is  the  union  of  two  or  more  individuala 
in  trade,  with  an  agreement  to  share  the  losses  and  profits 
in  the  ratio  of  the  amount  which  each  puts  into  the  partner- 
ship. The  money  employed  is  called  the  capital  stock.  The 
loss  or  gain  to  be  shared  is  called  the  dividend. 

A.,  B.,  and  C.  entered  into  copartnership.  A.  put  in  $180,  B.  put 
in  $240,  and  C.  put  in  $480.  They  gained  $300.  What  is  each  one's 
part  of  the  gain? 

$180  A.'s  stock-f  $240  B.'s  stock-f-$480  C.'s  stock=$900  wliole 
stock. 

£8°  =  *  =A.'s  part  of  the  entire  stock. 


Hence,  A.  must  have  £  of  $300=  $60  for  his  gain. 

B.  «        «    T\  of  $300=  $80    "     "      " 

C.  «        "    T8-jof$300=$160    "     "      " 

$300  verification. 
From  the  above  we  may  deduce  the  following 

RULE. 

Make  each  partner's  stock  the  numerator  of  a  fraction, 
and  the  sum  of  their  stock  a  common  denominator  ;  then 
multiply  the  whole  gain  or  loss  by  each  of  these  fractions,  for 
each  partner's  share. 

EXAMPLES. 

91.  A.  and  B.  purchase  a  house  for  $2500,  of  which  sum 
A.  furnished  $1200  and  B.  $1300.  They  receive  $210  rent 
for  the  same.  What  part  of  this  sum  ought  each  to  share  ? 

92-94.  A  person  failing  in  business  finds  that  all  his  debts 
amount  to  $4500,  and  that  he  has.  only  $2500  to  meet  these 


232  PROPORTION.  [CHAP.  xiv» 

claims.  How  much  ought  A.  to  receive,  whose  claim  is 
$360  ?  And  how  much  B.  whose  claim  is  $400  ?  And 
how  much  is  he  able  to  pay  on  the  dollar  ? 

95.  Two  brothers,  the  one  18  years  old  and  the  other  21 
years  old,  contribute  in  the  ratio  of  their  ages  $300  towards 
the  support  of  an  aged  parent.     What  did  each  contribute  ? 

96.  Two  persons,  A.  and  B.,  hire  a  pasture  for  $30,  into 
which  A.  turned  3  cows  and  B.  5.     What  part  of  the  $30 
ought  each  to  pay  ? 

97.  Five  persons,  A.,  B.,  C.,  D.,  and  E.,  are  to  share  be- 
tween them  $2400.     A.  is  to  have  l ;  B.  is  to  have  J  ;  C. 
is  to  have  -f  ;  D.  and  E.  are  to  divide  the  remainder  in  pro- 
portion to  the  numbers  5  and  7.     How  much  does  each  one 
receive  ? 

98.  There  are  three  horses  belonging  to  three  men,  em- 
ployed to  draw  a  load  of  plaster  a  certain  distance  for  $26'45. 
It  is  estimated  that  A.'s  and  B.'s  horses  do  f  of  the  labor  ; 
A.'s  and  C.'s  horses  T9^ ;  B.'s  and  C.'s  horses  if.     They 
are  to  be  paid  proportionally  according  to  these  estimates. 
What  ought  each  man  to  receive  ? 

99.  A.,  B.,  and  C.  agree  to  contribute  $365    towards 
building  a  church,  which  is  to  be  at  the  distance  of  2  miles 
from  A.,  2-J-  miles  from  B.,  and  3^  from  C.     They  agree 
that  their  shares  shall  be  proportional  to  the  reciprocals  of 
their  distances  from  the  church.     What  ought  each  to  con- 
tribute ? 

100.  A  person  wills  to  his  two  sons  and  a  daughter  the 
following  sums  :  to  the  elder  son  $1200,  to  the  younger  son 
$1000,  and  to  his  daughter  $600 ;  but  it  is  found  that  his 
whole  estate  amounts  to   only  $800.     How  much  ought 
each  child  to  receive  ? 

101.  Four  persons,  A.,  B.,  C.,  and  D.,  together  contribute 
$500  towards  the  erection  of  a  school-house,  which  is  placed 


§  127.]  DOUBLE  FELLOWSHIP.  233 

a,t  the  distance  of  -J  of  a  mile  from  A.'s  residence,  \  of  a  mile 
from  B.'s,  f  of  a  mile  from  C.'s,  and  1  mile  from  D.'s.  They 
contributed  in  the  reciprocal  ratio  of  their  respective  dis- 
tances from  the  school- house.  How  much  did  each  give  ? 


DOUBLE  FELLOWSHIP. 

§  127.  When  the  stock  of  the  several  partners  continues 
in  trade  for  unequal  periods  of  time,  the  profit  or  loss  must 
be  apportioned  with  reference  both  to  the  stock  and  time. 
In  such  cases  the  fellowship  is  called  DOUBLE  FELLOWSHIP. 

Three  partners,  A.,  B.,  and  C.,  put  into  trade  money  as  follows : 
A.  put  in  $400  for  2  months  ;  B.  put  in  $300  for  4  months ;  C.  put 
in  $500  for  3  months.  They  gained  $350.  How  must  they  share 
this  gain  ? 

It  is  evident  that  $400  for  2  months  is  the  same  as  $400X2=$800 
for  one  month;  $300  for  4  months  is  the  same  as  $300X4=$1200 
for  one  month;  $500  for  3  months  is  the  same  as  $500X3=$  1500 
for  one  month. 

Hence  $800  A.'s  money  for  one  month,-{-$1200  B.'s  money  for  one 
month,-f$1500  C.'s  money  for  one  month=$3500. 

Therefore,  by  Single  Fellowship, 

A.  must  have  ^fy=^g  of  $350=$80. 

B.  "        «      J|oo_i_3of    350=120. 
0.    "        "      £f£§=^  of   350=150. 

$350  verification. 

RULE. 

Multiply  each  partner's  stock  by  the  time  it  was  in  trade  ; 
make  each  product  the  numerator  of  a  fraction,  and  the  sum 
of  the  products  a  common  denominator ;  then  multiply  the 
whole  gain  or  loss  by  each  of  these  fractions,  for  each  part- 
ner s  share. 

20* 


234  PKOPOKTION.  [CHAP  xiv. 


EXAMPLES. 


102.  Two  persons,  A.  and  B.,  enter  into  partnership,  A. 
furnishing  $325  for  6  months,  and  B.  $200  for  8  months. 
What  ought  each  to  contribute  to  meet  a  loss  of  $100  ? 

103.  In  the  construction  of  a  piece  of  road,  A.  furnished 
5  laborers,  each  of  whom  worked  9  days  ;  B.  furnished  7 
laborers,  each  working  1 1  days :  for  the  whole  work  they 
received  $150.     What  was  each  one's  share  of  this  sum  ? 

104.  To  a  certain  school,  A.  sends  5  scholars  during  35 
days,  and  B.  sends  4  during  38  days,  and  has  to  pay  a  rate 
bill  of  $3-04.     What  was  A.'s  rate  bill  ? 

105.  If  I  borrow  $300  for  7  months  of  A.,  $400  for  8 
months  of  B.,  and  $450  for  9  months  of  C.  ;  for  this  ac- 
commodation I  wish  to  divide  $100  among  the  three.    What 
ought  each  to  receive  ? 

106.  For  the  transportation  of  100    barrels    of   flour  a 
distance  of  93  miles,  I  have  to  pay  $46 '50  to  5  individuals, 
who  performed  the  labor  as  follows  :  A.  carried  50  barrels 
a  distance  of  70  miles,  B.  carried  10  barrels  a  distance  of 
93  miles,  C.  carried  40  barrels  a  distance  of  53  miles,  D. 
carried  50  barrels  a  distance  of  23  miles,  and  E.  carried  40 
barrels  a  distance  of  40  miles.     How  much  ought  I  to  pay 
to  each  ? 

107.  Three  farmers  hired  a  pasture  for  $55*50  for  the 
season.     A.  put  in  6  cows  for  3  months,  B.  put  in  8  cows 
for  2  months,  C.  put  in  10  cows  for  4  months.     What  rent 
ought  each  to  pay  ? 

108.  On  the  first  day  of  January,  A.  began  business  with 
$650  ;  on  the  first  day  of  April  following,  he  took  B.  into 
partnership  with  $500  ;  on  the  first  day  of  next  July,  they 
took  in  C.  with  $450 :  at  the  end  of  the  year  they  found 
they  had  gained  $375.     What  share  of  the  gain  had  each  ? 


§  128.]  AVERAGE.  235 

109.  A.,  B.,  and  C.  have  together  performed  a  piece  of 
work  for  which  they  receive  $94f     A.  worked  12  days  of 
10  hours  each  ;  B.  worked  15  days  of  6  hours  each  ;  C. 
worked  9  days  of  8  hours  each.     How  ought  the  $94  to  be 
divided  between  them  ? 

110.  A  ship's  company  take  a  prize  of  $4440,  which  they 
agree  to  divide  among  them  according  to  their  pay  and  the 
time  they  have  been  on  board.     Now  the  officers  and  mid- 
shipmen have  been  on  board  6  months,  and  the  sailors  3 
months:  the  officers  have  $12  per  month,  the  midshipmen 
$8,  and  the  sailors  $6  per  month ;  moreover,  there  are  4 
officers,  12  midshipmen,  and  100  sailors.     What  will  each 
one's  share  be  ? 


CHAPTER    XV. 

AVERAGE. 

§  128.  IF  the  sum  of  a  series  of  unequal  quantities  be  di- 
vided by  the  number  of  quantities,  the  quotient  will  be  one 
of  a  series  of  equal  quantities,  whose  sum  will  equal  that  of 
the  former  series.  The  value  of  this  quotient  is  called  the 
AVERAGE  of  the  given  quantities.  Thus, 

A  laborer  worked  5  hours  on  Monday,  6  on  Tuesday,  3 
on  Wednesday,  9  on  Thursday,  9  on  Friday,  and  10  on  Sat- 
urday. How  many  hours  work  did  he  average  each  day  ? 
5  +  64-3  +  9  +  9+10=42;  and  42  hours -=- 6  =  Y  hours'  av- 
erage work  per  day.  Proof,  Yx6=:42.  Hence  the  fol- 
lowing rule  for  determining  the  average  : 

Divide  the  sum  of  the  given  quantities  by  the  number  of 
quantities.  The  quotient  will  be  the  average. 


236  AVEitAGh.  [CHAP.  xv. 


EXAMPLES. 

1-5.  What  is  the  average  of  1,  2,  3  ?  of  2,  3,  4,  7  ?  of 
5,  3,  4,  9,  4  ?  of  4,  9,  8,  7,  2,  6  ?  of  8,  4,  3,  9,  7,  12,  6  ? 

6-10.  Find  the  average  of  4,  7,  6,  2,  12  ;  of  12,  14,  19, 
18,  21  ;  of  6,  8,  13,  24,  30  ;  of  36,  42,  96,  104  ;  of  3,  4, 
5,  6,  7,  8,9,  10,  11,  12. 

11.  A  gentleman  expended  in  1845,  $1250'75  ;  in  1846, 
$1196-38;  in  1847,  $1341-67;  in  1848,  $1275'96  ;  in  1849, 
$1060-07  ;  in  1850,  $1196-27.  What  was  his  average  yearly 
expenditure  ? 

12.  The  following  was  the  record  of  attendance  for  one 
week  in  a  certain  school  :  Monday  A.  M.  109,  P.  M.  94  ; 
Tuesday  A.M.  109,  P.M.  103;    Wednesday  A.M.  97, 
P.  M.  91  ;  Thursday  A.  M.  104,  P.  M.  100  ;  Friday  A  M. 
88,  P.  M.  36.     What  was  the  average  half-day  attendance 
in  that  school  ? 

13.  Seven  men  weighed  each  as  follows  :  212/65.,  I35lbs., 
167J&S.,   196}Z6s.,   121JZ&S.,   102/fo.,   229^5.      What  was 
their  average  weight  ?     What  was  their  aggregate  weight  ? 

14.  What  was  the  average  cost  of  the  following  articles  ? 
The  first  cost  £2  45.  3d.  ;  the  2d,  £5   18s.  6d.  3qr.  ;  the 
3d,  £14  35.  2d.  ;  the  4th,  19s.  2qr. 

15.  Your  grandfather's  age  is  78  years  8  months;  your 
father's  is  54  years,  7  months,  22  days;  your  brother's  21 
years,  3  months,  29  days  ;  your  sister's  16  years  4  months  ; 
your  own  1  1  years,  6  months,  1  7  days.    What  is  the  average 
age  of  each  ? 

16.  At  sunrise  on  5  successive  days  the  barometer  was 
as  follows:    29'38  inches;  29'41  ;    29'63  ;    29'87  ;  30'06. 
What  was  the  average  height  of  the  mercury  during  this  time? 

17—18.  The  declination  of  the  sun  at  noon  on  the  first  7 
days  of  January,  1851,  was  as  follows:  23°  2'  14"  ;  22° 


§  128.]  AVERAGE.  237 

57' 10";  22°  51' 38";  22°  45'  39";  22°  39'  12";  22° 
32'  19"  ;  22°  24'  59".  What  was  the  average  declination 
of  the  sun  during  this  time  ?  At  the  same  times  respect- 
ively the  sun  was  slow  of  clock  3m.  44s.  ;  4m.  12s.  ;  4m. 
40s. ;  5m.  8s.  ;  5m.  35s. ;  6m.  2s.  ;  6m.  28s.  What  was 
the  average  time  which  the  sun  was  slow  of  clock  ? 

19.  By  observation  the  length  of  a  pendulum  vibrating 
once  in  a  second  of  time,  is  found  to  be,  at  the  equator, 
39-01612  inches;    at  the  Cape  of  Good  Hope  39'07815 
inches  ;  at  New  York  39-10120  inches  ;  at  Paris  39-12929  ; 
at  London  39*13929  inches.     What  is  the  average  length 
for  these  five  places  ? 

20.  The  mean  distances  of  the  four  satellites  of  Jupiter 
are  as  follows,  the  radius  of  Jupiter  being  taken  as  the  unit : 
6-04853  ;  9'62347  ;   15'35024  ;  26'99835.     What  is  their 
average  mean  distance  ? 

21-22.  A  locomotive  made  7  successive  trips  over  a  track 
of  17  miles  in  the  following  times:  50m.  32s.  ;  49m.  3s. ; 
48771.  10s. ;  40m.  30s. ;  41m.  35s. ;  45m.  45s. ;  44m.  20s. 
What  was  the  average  time  of  one  trip  ?  What  was  the 
average  time  of  running  one  mile  ? 

23.  A  company  of  6  California  gold  diggers  find  on  a  cer- 
tain day  gold  as  follows:  the  1st,  702.  I3pwt. ;  the  2d,  9oz. 
I4pwt. ;  the  3d,  6oz.  IQpwt.;  the  4th,  4oz.  4pwt. ;  the  5th, 
lOoz.  8pwt.  ;  the  6th,  3oz.  2pwt.  What  was  the  average 
for  each  man  ? 

24-25.  On  a  certain  day  in  January  I  noticed  the  ther- 
mometer to  be  as  follows  :  at  6  A.  M.  20°  ;  at  7,  23°  ;  at 
8,  25°  ;  at  9,  30° ;  at  10,  36°  ;  at  11,  40o  ;  at  noon,  44° ; 
at  1  P.  M.  45°  ;  at  2,  48°  ;  at  3,  46°  ;  at  4,  44°  ;  at  5,  39°  ; 
at  6,  33°.  What  was  the  average  during  the  forenoon, 
and  what  during  the  afternoon,  if  the  observation  at  noon 
is  not  taken  into  the  account  ? 


238  AVERAGE.  [CHAr. 


EQUATION  OF  PAYMENTS. 

§  129.  EQUATION  OF  PAYMENTS  is  a  process  by  which  we 
ascertain  the  average  time  for  the  payment  of  several  sums 
due  at  different  times. 

Suppose  I  owe  $1000,  of  which  $100  is  due  in  2  months,  $250  in 
4  months,  $350  in  6  months,  and  $300  in  9  months.  If  I  pay  the 
whole  sum  at  once,  how  many  months'  credit  ought  I  to  have  ? 

A  credit  on  $100  for  2  months  is  the  same  )  n 

J-2»io.XloO=  2007WO. 
as  a  credit  on  $1  for  200  months.  ) 

A  credit  on  $250  for  4  months  is  the  same 
as  a  credit  on  $1  for  1000  months. 

A  credit  on  $350  for  6  months  is  the  same  )  „ 
as  a  credit  on  $1  for  $2100  months.  [  6»«*X850=2100mo. 

A  credit  on  $300  for  S >  months  is  the  same  )  = 

as  a  credit  on  $1  for  2700  months.  f         

$1000     6000wo. 

Hence,  I  ought  to  have  the  same  as  a  credit  on  $1  for  6000  months. 
But  if  I  wish  a  credit  on  $1000  instead  of  $1,  it  ought  evidently  to 
be  for  only  one-thousandth  part  of  6000  months,  which  is  6  months. 

Hence  this 

RULE. 

Multiply  each  sum  by  the  time  that  must  elapse^before  it 
becomes  due ;  divide  the  amount  of  these  products  by  the 
amount^  of  the  sums  ;  the  quotient  will  be  the  equated  time. 

EXAMPLES. 

26.  I  purchased  a  bill  of  goods  amounting  to  $1500,  of 
which  I  am  to  pay  $300  in  2  months,  $500  in  4  months,  and 
the  balance  in  6  months.     What  would  be  the  mean  time 
for  the  payment  of  the  whole  ? 

27.  A  merchant  owes  $500  to  be  paid  in  6  months,  $600 
to  be  paid  in  8  months,  and  $400  to  be  paid  in  12  months. 
What  is  the  average  time  of  payment  ? 


§  129.]  EQUATION  OF  PAYMENTS.  239 

28.  A.  owes  B.  a  certain  sum  :    one- third  is  due  in  6 
months,  one-fourth  in  8  months,  and  the  remainder  in   1 2 
months.     What  is  the  mean  time  of  payment  ? 

NOTE. — It  makes  no  difference  what  the  amount  is  which  A.  owes 
B.,  since  it  is  certain  fractional  parts  which  becomes  due  at  particular 
times.  If  we  suppose  the  sum  to  be  $1,  then  our  work  will  be 

mo.  mo. 
$  X   6  =  2 
4  X   8  =  2 
Remainder  is  y5^,  and  y5^Xl2  =  5 

29.  A  merchant  has  due  him  $300  to  be  paid  in  2  months, 
$800  to  be  paid  in  5  months,  $400  to  be  paid  in  10  months. 
What  is  the  average  time  for  the  payment  of  the  whole  ? 

30.  A  merchant  owes  $1200,  payable  as  follows:  $200 
in  2   months,  $400  in  5  months,  and  the  remainder  in  8 
months.     He  wishes  to  pay  the  whole  at  one  time.     What 
is  the  average  time  of  such  payment  ? 

31.  A  merchant  bought  goods  to  the  amount  of  $2400, 
for  one-fourth  of  which  he  was  to  pay  cash  at  the  time  of 
receiving  the  goods,  one-third  in  6  months,  and  the  balance 
in  10  months.     What  was  the  equitable  time  for  the  pay- 
ment of  the  whole  ? 

32.  Suppose  I  owe  $100  payable  on  January  1st,  $150 
on  February  5th,  $300  on  April  10th.     If  we  count  from 
January  1st,  and  allow  29  days  to  February,  it  being  leap 
year,  on  what  day  ought  the  whole  sum  in  equity  to  be 
paid? 

NOTE. — Estimate  the  time  in  days.  The  1st  payment  is  $100  due 
in  0  days. 

33.  A  merchant  bought  a  bill  of  goods  amounting  to  $1000. 
He  agreed  to  pay  $250  the  first  day  of  the  next  March,  $250 
on  the  3d  of  the  following  May,  $250  on  the  4th  of  the 
following  July,  and  the  remaining  $250  on  the  15th  of  the 


240  AVERAGE.  [CHAP.  A 

following  September.     What  would  be  the  equitable  time 
for  paying  the  whole  ? 

NOTE. — As  the  sums  are  equal,  it  will  simplify  the  operations  tc 
consider  each  payment  $1. 

34.  A  person  purchased  a  bill  of  goods  amounting  to 
$3450,  and  agreed  to  pay  as  follows :  $1000  at  the  end  of 
3  months,  $1000  at  the  end  of  6  months,  and  the  balance 
at  the  end  of  9  months.     What  was  the  average  time  for 
which  he  received  credit  on  the  whole  sum  ? 

35.  A  person  owes  as  follows  :  $300  due  the  10th  of 
March,  $250  due  the  28th  of  March,  $450  due  the  31st 
of  March,  and  $100  due  the  25th  of  the  following  April. 
At  what  time  could  the  whole  sum  in  equity  be  paid  ? 

36.  A  person  owes  a  certain  sum  of  money,  ^  of  which 
is  due  in  3^  months,  J  is  due  in  4J  months,  J  is  due  in  5 
months,  and  the  balance  is  due  in  8  months.     What  is  the 
mean  time  of  payment  ? 

37-38.  A  person  purchases  a  farm  for  $7000,  and  agrees 
to  pay  as  follows  :  $1000  at  the  end  of  3  months  ;  $1500 
at  the  end  of  4  months ;  $2000  at  the  end  of  5  months ; 
$2500  at  the  end  of  6  months.  At  what  time  in  equity 
ought  he  to  pay  the  whole  ?  Suppose  he  had  agreed  to 
pay  $2500  at  the  end  of  3  months,  $2000  at  the  end  of  4 
months,  $1500  at  the  end  of  5  months,  and  $1000  at  the 
end  of  6  months  ;  then,  in  equity,  at  what  time  ought  the 
whole  to  be  paid  ? 

39.  A  sum  of  money  is  due  as  follows:  -i  on  the  1st  of 
July,  }  on  the  1st  of  August,  -J  on  the  1st  of  September, 
jJg-  on  the  1st  of  October,  and  the  balance  on  the  1st  of  No- 
vember. At  what  time,  estimating  from  the  1st  of  July, 
ought  the  whole  in  equity  to  be  paid  ? 

§  130.  Suppose  $1000  to  be  due  at  the  end  of  6  months; 


§  130.]  EQUATION  OF  PAYMENTS.  241 

that  3  months  before  it  is  due  $100  was  paid,  and  that  1 
month  before  the  expiration  of  the  6  months  $300  was  paid. 
How  long  after  the  end  of  the  6  months  may  the  balance 
of  $600  remain  unpaid  ? 

NOTE. — The  problem  here  is,  when  a  debt  due  at  some  future  pe- 
riod has  received  several  partial  payments  before  the  time  due,  to 
find  how  long  beyond  this  time  the  balance  may  in  equity  remain 
unpaid. 

3wo.XlOO=300wo. ;  lmo.X300=300/«o. ;  that  is,  $1  must  have 
a  credit  of  300mo.+300mo.=600mo.  The  balance  due  is  §600,  which 
must  have  a  credit  equal  to  600/«o.-7-600=lmo.  beyond  the  6  months. 

Hence  this 

RULE. 

Multiply  each  payment  by  the  time  it  was  paid  before  due  ; 
then  divide  the  sum  of  the  products  thus  obtained  by  the  bal- 
ance remaining  unpaid  ;  the  quotient  will  be  the  equated  time. 


EXAMPLES. 

40.  Suppose  $1496-41  to  be  due  at  the  end  of  90  days : 
that  84  days  before  it  is  due  there  is  paid  $500 ;  32  days 
before  the  90  days  expire  there  is  paid  $502'50.     How  long 
after  the  90  days  before  the  balance  of  $493 '91  ought  to 
be  paid  ? 

41.  A.  lent  $200  to  B.  for  8  months ;  at  another  time  he 
lent  him  $300  for  6  months.     For  how  long  a  time  ought 
B.  to  lend  A.  $800  to  balance  the  favor  ? 

42.  A  person  owes  $1000,  due  at  the  end  of  12  months. 
At  the  end  of  3  months  he  pays  $100,  one  month  after  that 
he  pays  $100.     How  long  beyond  the  12  months  may  the 
balance  of  $800  remain  unpaid  ? 

43-44.  A  credit  of  6  months  on  $500,  and  of  4  months 
on  $1000,  is  the  same  as  a  credit  of  how  many  dollars  for 

21 


24:2  AVERAGE.  [CHAP,  xv 

8  months  ?     It  is  the  same  as  a  credit  on  $800  for  how  many 
months  ? 

§131.  It  is  customary  with  many  merchants  to  give  a 
credit  of  from  3  to  6  months,  on  their  bills  of  sale.  In 
such  cases,  in  settling  up  their  accounts,  which  generally 
consist  of  various  items  of  debit  and  credit  at  sundry  times, 
it  is  very  desirable  to  have  some  simple  rule  by  which  the 
cash  balance  can  be  found. 

Suppose  A.  has  the  following  account  with  B.  : 
1848.  Dr.      i       1848.  (  r. 

Jan.     10.    To  Merchandise    .    .    $  100     Feb.     8.    By  Merchandise  ...    $50 
March  2G.      "  "  .    .      400  I  April  23.     «•  "  ...    375 

What  is  the  cash  balance,  July  10,  1848,  if  interest  is  estimated 
at  7  per  cent.,  and  a  credit  of  30  days  is  allowed  on  all  the  different 
sums  ? 

If  interest  were  not  considered,  the  above  account  could  be  bal- 
anced as  follows : 


1848.  Dr. 

Jan.      10.    To  Merchandise    .    .   $100 
March  20.     «  "  400 


$500 


1848.  Cr. 

Feb.     8.    By  Merchandise  .    .    .    $50 
April  23.     «  "  ...    375 

"   Balance      ....      75 

$500 


To  Balance    $75 

Had  no  credit  been  given,  the  debits  should  be  increased  by  the 
following  items  of  interest.  (§  83,  note  4,  and  §  105.) 

On  $100  for  182  days,  at  7  per  cent.=100X182X|^7 
"     400   "   106     "  "         "        =400X106X^°T- 

In  like  manner  the  credits  should  be  increased  by  interest : 
On  $50  for  153  days,  at  7  per  cent.=  50X153X|^7. 
"    375    "     78     "          «        «        =375  X  78  X^7. 
But,  since  30  days'  credit  is  given  on  all  sums,  it  follows  that  by  the 
above,  we  should  increase  the  debits  by  an  excess  of  interest  equal 
to  the  interest  of  the  sum  of  debits,  $500  for  30  days=500  X  30  X 
f^5 .     In  like  manner  we  should  increase  the  credits  by  an  excess  of 
interest  equal  to  the  interest  of  sum  of  credits,  §425,  for  30  days= 


§  131.]  EQUATION  OF  PAYMENTS.  243 

Now  if,  instead  of  diminishing  the  debit  items  of  interest  by  500 
X30Xfff°37,  and  the  credit  items  of  interest  by  425X30X  |^7,  we 
merely  diminish  the  debit  items  of  interest  by  the  interest  on  mer- 
chandise balance,  $75  for  30  days,  which  is  75X30X§VVi tlie  resulfc 
will  be  the  same.  And  since  taking  any  sum  from  one  side  of  a  book 
account  has  the  same  effect  as  adding  the  same  sum  to  the  other 
side,  it  follows,  that  instead  of  diminishing  the  debit  items  of  interest 
by  75X30Xj^37,  we  may  increase  the  credit  items  of  interest  by 
this  same  quantity. 

From  which  we  see  that  the  difference  between  1 00X182  X§^j7+ 
400X106X§^7  and  50Xl53X^7+375X78Xf¥y+75X30XfV°3 
is  the  interest  balance. 

The  operations  indicated  in  the  foregoing  work  may  be  exhibited 
in  a  more  condensed  form,  as  follows  : 

DEBITS.  CREDITS. 

$        Days.  $        Days. 

100X182=18200  50X153=  7650 

400X106=42400  375  X   78=29250 

75  X   30=  2250 


60600 

39150  39150 


2U50=$4-ll=mteres*  balance. 
Hence  the  foregoing  account  will  become  balanced  as  follows : 


1848.  Dr. 

Jan.      10.  To  Merchandise    .    $100-00 

March  26.  "  **  •      400-00 

July     10.  "   balance  of  interest     4-11 

$504-11 

July     10.     "    Cash  balance  .    .  $79-11 
From  the  above,  we  deduce  this 


1848.  Cr. 

Feb.     8.  By  Merchandise   .    .  $50-00 

April  23.  «             "             .    .  375-00 

July    10.  "    balance  ....  79-11 

$504-11 


RULE. 

Place  such  sum  on  the  debtor  or  credit  side  as  may  be 
necessary  to  balance  the  account,  which  sum  may  be  regarded 
as  MERCHANDISE  BALANCE.  Then  multiply  the  number  of 
dollars  in  each  entry  by  the  number  of  days  from  the  time 
such  entry  was  made,  to  the  time  of  settlement  ;  observing  to 


AVERAGE. 


[CHAP.  xv. 


multiply  the  merchandise  balance  by  the  number  of  days  for 
which  credit  is  given. 

Multiply  the  difference  between  the  sum  of  the  debit  prod- 
ucts, and  the  sum  of  the  credit  products,  by  the  interest  of 
$1  for  1  day  ;  the  product  will  be  the  number  of  dollars  in 
INTEREST  BALANCE,  which  ivill  be  in  favor  of  the  debit  side  of 
account,  when  the  sum  of  debit  products  exceeds  the  sum  of 
credit  products  ;  but  in  favor  of  the  credit  side  when  the 
sum  of  credit  products  exceeds  the  sum  of  debit  products.  If 
then,  the  interest  balance  be  added  to,  or  subtracted  from,  the 
merchandise  balance,  as  the  case  may  require,  it  will  give  the 
cash  balance. 

EXAMPLES. 

45.  Suppose  A.  has  the  following  account  with  B.  : 


1848. 

Jan.      1.  To  Merchandise  . 

March  3.  "  « 

May    10.  «  " 

June    6.  "  « 


$200 
500 
100 
300 

1100 
.950 


1848. 

Jan.      15.    By  Merchandise 
March  20.     "  ** 


May 
July 


Cr. 

$300 

400 

200 

50 

$950 


Merchandise  balance    $150 

What  is  the  cash  balance  of  the  above  account  on  the 
1st  of  July,  1848,  provided  each  individual  is  allowed  90 
days'  time  on  his  purchases,  if  interest  is  estimated  at  7  per 
cent.  ? 

NOTE. — The  interest  balance  will  be  found  in  favor  of  the  credit 
side ;  the  merchandise  balance  is  in  favor  of  the  debtor's  side. 

46-47.  A.  has  the  following  account  with  B  : 


1850. 

March  9.  To  Merchandise 
April    4.     «          « 
June   12.     « 
July    17.     "          " 


Dr. 
$18-38 
56-41 
105-03 

88-13 

$267-95 


1850. 

March  28.  By  Merchandise 
July       2.     «  ** 

"        30.      "  " 

Aug.    20.      «  « 


Cr. 

$60-20 

100-00 

2C3-40 

75-75 

499-35 
267-95 


Merchandise  balance    $231-40 


§  131.] 


EQUATION  OF  PAYMENTS. 


245 


What  was  the  cash  balance  of  the  above  account,  and  in 
whose  favor,  on  the  1st  day  of  October,  1850,  provided  each 
individual  is  allowed  90  days'  time  on  his  purchases,  interest 
being  6  per  cent.  ? 

What  was  the  cash  balance  of  the  above,  on  the  1st  day 
of  January,  1851,  the  other  conditions  remaining  the  same  ? 

48.  Suppose  A'.'s  account  with  B.  to  have  been  as  fol- 
lows : 


1848. 
Jan.    10. 
Feb.  25. 
March  1. 


To  Merchandise 


Dr. 

$250-37 
113-04 
405-59 

1848. 
June  25. 
July  20. 
July  28. 

By  Merchandise 

t;              u 
it              tc 

769-00 

688-52 

$80-48 

Cr. 
$37-51 

50-98 
60003 

$688-52 


What  is  the  cash  balance,  and  in  whose  favor,  on  the  1st 
of  August,  1848,  provided  6  months,  or  180  days'  time  is 
given,' interest  being  6  per  cent.  ? 

NOTE. — In  practice,  when  the  cents  Jn  any  of  the  entries,  as  in  this 
example,  are  less  than  50,  we  may,  without  sensible  error,  omit  them ; 
but  when  they  are  50,  or  greater,  we  may  consider  them  as  an  ad- 
ditional dollar. 

49-50.  A.'s  account  with  B.  is  as  follows  • 


1850. 


Dr. 


September  2.    To  Merchandise,  $212-14 


25. 

October  24. 
Novemb'r21. 
Decerab'r  24. 


405-21 
303-60 
140-80 
28-30 


1090-05 
916-92 


1850. 


Cr. 


September  13.    By  Merchandise,  $300  00 


October 


30. 

28. 


212-12 
404-80 

$916-92 


Merchandise  balance    $173-13 

What  was  the  cash  balance,  and  in  whose  favor,  of  the 
above  account,  on  the  1st  day  of  January,  1851,  if  each  in- 
dividual had  a  credit  of  4  months  or  120  days,  interest  being 
7  per  cent.  ? 

21*. 


246  AVERAGE.  [CHAP.  xv. 

What  was  the  cash  balance  on  the  21st  day  of  June,  1851, 
the  other  conditions  remaining  the  same  ? 


ALLIGATION  MEDIAL. 

§  132.  ALLIGATION  MEDIAL  teaches  the  method  of  finding 
the  average  or  mean  value  of  a  compound,  when  its  several 
ingredients  and  their  respective  values  are  given. 

A  grocer  mixes  140  pounds  of  tea,  worth  8s.  per  pound;  200 
pounds,  worth  6s.  per  pound  ;  and  160  pounds,  worth  10s.  per  pound. 
What  is  a  pound  of  the  mixture  worth  ? 

140  pounds  of  tea,  at  8s.  per  pound,  are  worth  140X8=1120s.  , 
200  pounds,  at  6s.,  are  worth  200X6=1200s.  ;  160  pounds,  at  10s., 
are  worth  160XlO=1600s.  Therefore,  the  mixture,  which  is  500 
pounds,  is  worth  1120+1200+1600=3920s.  ;  and  one  pound  must 
he  worth  3A2o°^!^ 

Hence,  to  find  the  mean  value  of  a  compound,  composed  of  several 
ingredients  of  different  values,  we  have  this 


Divide  the  sum  of  the  values  of  all  the  quantities  by  the 
sum  of  the  quantities. 

EXAMPLES. 

51.  A  wine  merchant  mixed  several  sorts  of  wine,  viz: 
32  gallons,  at  40  cents  per  gallon  ;   15  gallons,  at  60  cents 
per  gallon  ;  45  gallons,  at  48  cents  per  gallon  ;  and  8  gal- 
lons, at  85  cents  per  gallon.     What  is  the  value  of  a  gallon 
of  the  mixture  ? 

52.  A  farmer  mixed  together  7  bushels  of  rye,  worth  72 
cents  per  bushel  ;  15  bushels  of  corn,  worth  60  cents  per 
bushel;  and  12  bushels  of  wheat,  worth  $1*20  per  bushel. 
What  is  the  value  of  a  bushel  of  the  mixture  ? 

53.  A  goldsmith  melts  together  11    ounces  of  gold  23 


§  132.]  ALLIGATION  MEDIAL.  247 

carats  fine,  8  ounces  21  carats  fine,  10  ounces  of  pure  gold, 
and  2  pounds  of  alloy.    How  many  carats  fine  is  the  mixture  ? 
NOTE. — A  carat  is  a  24th  part.     Thus,  21  carats  fine  is  the  same 
as  |]  pure  metal. 

54.  On  a  certain  day,  the  mercury  in  the  thermometer 
was  observed  to  stand  2  hours  at  62  degrees,  4  hours  at  70 
degrees,  5  hours  at  72  degrees,  3  hours  at  59  degrees,  and 
1  hour  at  75  degrees.     What  was  the  mean  temperature 
for  the  15  hours  ? 

55.  Suppose  a  ship  sail  at  the  rate  of  5  knots  for  3  hours, 
at  7  knots  for  5  hours,  and  8  knots  for  4  hours.     What  is 
her  rate  of  sailing  during  the  1 2  hours  ? 

56.  A  grocer  mixes  30  pounds  of  sugar,  worth  10  cents 
per  pound;  40  pounds,  worth  10^  cents  per  pound;  24 
pounds,  worth  11  cents  per  pound;  and  60  pounds,  worth 
13  cents  per  pound.    What  is  a  pound  of  the  mixture  worth  ? 

57.  A  person  bought  4  dozen  of  eggs,  at  18J  cents  per 
dozen;  6  dozen,  at  21  cents  per  dozen  ;  3£  dozen,  at  24  cts. 
per  dozen  ;  5  J  dozen,  at  25  cents  per  dozen.     What  was 
the  average  cost  of  one  dozen  ? 

58.  A  flour  merchant  bought  300  barrels  of  flour,  at  $3 '75 
per  barrel ;  250  barrels,  at  |3'87i  per  barrel;  500  barrels, 
at  $3'93-j  per  barrel.     What  did  the  whole  average  per 
barrel  ? 

59.  A  dairyman  made  during  the  first  month,  26  cheeses, 
each  weighing  85  pounds;  during  the  second  month,  he 
made  25,  each  weighing  83  pounds ;  and  during  the  third 
month  he  made  20,  each  weighing  80£  pounds.     What  was 
the  average  weight  of  his  cheese  for  the  3  months  ? 

60.  A  dairyman  having  30  cows,  finds  at  a  certain  milking 
that  6  give  12  quarts  of  milk  each  ;  8  give  10^  quarts  each  ; 
10  give  9j  quarts  each ;  and  the  others  give  only  8  quarts 

What  did  each  cow  on  an  average  give  ? 
G 


248  AVEKAUE.  [CHAP.  xv. 

NOTE. — It  will  be  seen  that  the  principle  of  Equation  of  Payments 
and  that  of  Alligation  Medial  are  the  same :  in  the  one  case,  we  oper- 
ate upon  debts,  and  payments,  and  time  ;  in  the  other,  upon  ingre- 
dients or  quantities  and  values. 


ALLIGATION  ALTERNATE. 

§  13$.  ALLIGATION  ALTERNATE  is  the  reverse  of  Alligation 
Medial ;  that  is,  it  teaches  the  method  of  determining  the 
proportional  quantities  of  several  ingredients,  so  that  the 
compound  shall  have  a  given  value. 

Suppose  we  wish  to  mix  teas,  which  are  worth  4  and  6  shillings  per 
pound,  so  that  the  mixture  may  be  worth  5  shillings  per  pound  ;  it 
is  obvious  that  we  must  take  equal  quantities  of  each,  since  the  price 
of  the  one  is  as  much  less  than  the  average  price,  as  the  other  is 
greater. 

Again,  suppose  we  wish  to  mix  teas  which  are  worth  4  and  7  shil- 
lings per  pound,  so  that  the  mixture  may  be  worth  5  shillings.  In 
this  case  the  7  shilling  tea  is  2  shillings  above  the  average  price, 
whilst  the  4  shilling  tea  is  but  1  shilling  below.  It  will  be  necessary 
to  use  twice  as  much  of  the  4  shilling  tea  as  of  the  7  shilling  tea ;  and 
in  all  cases  it  is  obvious  that  the  quantities  to  be  used  will  be  in  the 
inverse  ratio  to  the  differences  between  their  prices  and  the  mean  price. 

When  there  are  more  than  two  simples  they  may  be  compared 
together  in  couplets,  one  term  of  which  must  obviously  exceed  the 
average  price,  while  the  other  must  be  less. 

CASE  I. 

The  rates  of  the  several  ingredients  being  given,  to  make 
a  compound  of  a  fixed  rate. 

RULE. 

I.  Write  the  rates  of  the  simples  in  a  vertical  column. 
Connect  the  rate  of  each  ingredient  which  is  less  than  the 
rate  of  the  compound,  with  one  or  more  rates  greater  than  the 
rale  of  the  compound ;  connect  in  the  same  icay,  each  rate 


§  133.]  ALLIGATION   ALTERNATE.  249 

which  is  greater  than  the  rate  of  the  confound,  with  one  or 
more  rates  which  are  less. 

II.  Write  the  difference  between  the  rate  of  each  one  ingre- 
dient and  the  value  of  the  compound,  opposite  the  rate  of 
each  other  ingredient  with  which  the  former  is  connected.  If 
only  one  difference  stands  against  any  rate,  it  will  be  the  re- 
quired quantity  of  the  ingredient  of  that  rate  ;  but  if  there 
be  several,  their  sum  will  be  the  quantity  required. 

How  much  sugar  at  5,  6,  and  10  cents  per  pound,  must 
be  mixed  together,  so  that  a  pound  of  the  mixture  may  be 

worth  8  cents  ? 

5 


3+2=5 

Therefore,  if  we  take  2  pounds  at  5  cents,  2  pounds  at  6  cents, 
and  5  pounds  at  10  cents,  we  shall  satisfy  the  conditions  of  the  ques- 
tion. It  is  obvious,  that  any  other  quantities  of  the  several  ingre- 
dients which  are  to  each  other  as  the  numbers  2,  2,  and  5,  will  satisfy 
the  question  equally  well ;  so  that  in  Alligation  Alternate  the  num- 
ber of  solutions  are  indefinite  ;  all  that  we  can  do  is  to  find  the  ratios 
of  the  quantities  required. 

In  many  cases  the  ingredients  will  admit  of  being  connected  in 
several  ways,  and  then  we  shall  obtain  as  many  sets  of  ratios  as  there 
are  methods  of  connecting  them ;  for  example  : 

How  many  pounds  of  raisins  at  4,  6,  8,  and  10  cents  per  pound, 
must  be  mixed,  so  that  a  pound  of  the  compound  may  be  worth  1 
cents  ? 

In  this  question  the  terms  may  be  connected  in  seven  distinct 
ways ;  therefore,  we  shall  obtain  seven  sets  of  ratios,  as  follows  : 


250  AVEKAUE.  [CHAP.  xv. 

How  much  tea  at  5  shillings,  6  shillings,  and  8  shillings  per  pound, 
must  be  mixed,  so  that  the  mixture  may  be  worth  7  shillings  per 
pound  ? 

If  we  compound  only  the  5  and  8  shilling  teas,  we  must  take  them 
in  the  ratio  of  1  to  2,  since  7  shillings  is  1  shilling  less  than  8  shillings, 
and  2  shillings  greater  than  5  shillings.  Hence,  any  one  of  the  com- 
pounds in  the  following  group  (A)  will  be  worth  7  shillings  per  pound. 

(1)    (2)     (3)    (4)    (5)    (6)        1 

5  shilling  tea     1       2       3       4       5       6,  Ac.   I    ... 
8  shilling  tea     2       4       6       8     10     12,  Ac.   |l    ' 

Sums,     3  ;     6  ;     9  ;  12  ;  15  ;  18,  Ac.  j 

If  we  now  mix  the  6  and  8  shilling  teas,  we  see  that  it  will  be  ne- 
cessary to  take  equal  quantities  of  each,  since  the  average  price  is  to 
be  as  much  above  6  shillings  as  it  is  belov  8  shillings.  Hence,  the 
following  compound  will  also  be  worth  7  shillings  per  pound. 

(1)    (2)    (3)    (4)    (5)    (6) 

6  shilling  tea     1       2       3       4       5       6,  Ac. 
8  shilling  tea     1       2       3       4       5       6,  Ac. 

Sums     2  ;     4  ;     6  ;     8  ;  10  ;  12,  Ac.  J 

Now,  it  is  obvious,  we  may  combine  any  one  of  these  last  results 
with  any  one  of  the  former  results.  Thus,  if  we  combine  (1)  of  group 
(A)  with  (1)  of  (B),  we  have 

Pounds. 

5  shilling  tea 1 

6  "         "  1 

8        "         "  2+1=3 

If  we  combine  (1)  of  (A)  with  (2)  of  (B),  we  have 

Pounds. 

5  shilling  tea 1 

6  "         "  2 

8        "         "  2+2=4 

Combining  (2)  of  (A)  with  (3)  of  (B),  we  have 

Pounds. 

5  shilling  tea 2 

6  "         "  3 

8        «         «  4+3=7 


§  133.]  ALLIGATION   ALTERNATE.  251 

Combining  (5)  of  (A)  with  (4)  of  (B),  we  have 

Pounds. 

5  shilling  tea 5 

6  "         "  4 

8        *        "  10-f-4=14 

The  number  of  combinations  which  could  be  made  in  this  way  ia 
unlimited;  hence  the  above  class  of  questions  in  Alligation  admit 
of  an  infinite  number  of  answers. 

EXAMPLES. 

61-66.  How  much  wine,  at  $1-12  per  gallon  and  48  cents 
per  gallon,  must  be  mixed  together,  that  the  composition 
may  be  worth  60  cents  per  gallon  ?  65  cts.  ?  72  cts.  ?  84 
cts.?  91  cts.?  $1-02? 

67.  How  many  gallons  of  wine  and  water  must  be  mixed 
together,  that  the  mixture  may  be  worth  60  cents  per  gal- 
lon, the  water  being  considered  of  no  value,  and  the  wine 
with  which  it  is  mixed  being  worth  90  cents  per  gallon? 

68-71.  Having  gold  of  12,  16,  17,  and  22  carats  fine, 
what  proportion  of  each  kind  must  I  take,  to  make  a  com- 
pound of  18  carats  fine  ?  19  ?  20  ?  21  ? 

72-76.  How  much  of  each  sort  of  grain,  at  56,  62,  and 
75  cents  per  bushel,  must  be  taken,  so  that  the  mixture  may 
be  worth  60  cents  per  bushel  ?  65  cts.  ?  68  cts.  ?  70  cts.  ? 
72  cts.  ? 

77-81,  How  much  tea  at  4  shillings,  5  shillings,  6  shillings, 
and  12  shillings  per  pound,  must  be  mixed  that  the  mixture 
may  be  worth  7  shillings  per  pound  ?  8s.  ?  95.  ?  105.  ?  115.  ? 

CASE    II. 

When  one  of  the  ingredients  is  limited  to  a  certain  quantity. 

A  person  wishes  to  mix  10  bushels  of  wheat,  worth  $1  per  bushel, 
with  rye,  worth  70  cents  per  bushel,  and  oats,  worth  30  cents  per 


252  AVERAGE.  [CHAP.  xv. 

bushel,  so  that  the.  mixture  may  be  worth  60  cents  per  bushel.    How 
many  bushels  of  r\  e  and  oats  must  he  use  ? 

Proceeding,  according  to  Case  I.,  we  find  the  proportionate  numbers 
to  be  30,  30,  and  50.     Hence, 

30  :  30  :  :  10  :  10 
30  :  50  :  :  10  :  16§ 

So  that  he  must  make  use  of  10  bushels  of  rye  and  16§  bushels 
of  oats.    Hence,  this 


RULE. 


Find  ike  proportionate  quantities  of  each  ingredient,  by 
Case  I.,  as  though  there  was  no  limitation  ;  then,  as  the  dif- 
ference against  the  simple  whose  quantity  is  given,  is  to  each 
of  the  other  differences,  so  is  the  given  quantity  of  that  simple 
to  the  quantity  required  of  each  of  the  other  simples. 


EXAMPLES. 


82-85.  A  grocer  has  90  pounds  of  tea,  worth  90  cents 
per  pound,  which  he  wishes  to  mix  with  three  other  quali- 
ties, valued  at  80  cents,  70  cents,  and  60  cents  per  pound. 
How  much  must  he  take  of  these  three  kinds,  so  as  to  be 
able  to  sell  the  mixture  at  65  cents  per  pound  ?  at  68  cts.  ? 
at  85  cts.?  at  8 7 Jets.? 

86-91.  A  merchant  has  90  pounds  of  spice  worth  86 
cents  per  pound,  which  he  wishes  to  mix  with  three  other 
sorts  which  are  worth  30,  40,  and  50  cents  per  pound,  re- 
spectively. How  many  pounds  must  be  used  so  that  the 
compound  may  be  worth  55  cents  per  pound  ?  60  cts.  ?  65 
cts.  ?  70  cts.  ?  75  cts.  ?  80  cts.  ? 

92-96.  A  merchant  wishes  to  mix  100  pounds  of  sugar, 
worth  10  cents  per  pound,  with  three  other  kinds  worth  9, 
8,  and  5  cents  per  pound,  respectively.  How  many  pounds 
must  he  use  so  that  the  compound  may  be  worth  5  J  cts.  ? 
6  cts.  ?  6J  cts.  ?  7  cts.  ?  7|  cts.  ? 


$134.] 


INVOLUTION. 


253 


CHAPTER    XVI. 

INVOLUTION  AND  EVOLUTION. 

INVOLUTION. 

§134.  THE  product  arising  from  multiplying  a  number 
into  itself  is  called  the  second  power,  or  the  square  of  that 
number.  Thus,  3x3=9:  9  is  the  square  of  3.  If  the 
square  of  a  number  be  again  multiplied  by  that  number,  the 
result  is  called  the  third  power,  or  the  cube  of  the  number. 
Thus,  3x3x3  =  27:  the  number  27  is  the  cube  of  3. 

The  word  power  denotes  the  product  arising  from  multi- 
plying a  number  into  itself  a  certain  number  of  times ;  and 
the  number  thus  multiplied  is  called  the  root.  Thus,  9  is 
the  second  power  of  3,  and  3  is  the  square  root  of  9.  In 
the  same  manner  27  is  the  third  power  of  3,  and  3  is  the 
cube  root  of  27. 
Involution  is  the  process  of  raising  a  number  to  a  given  power. 

To  denote  that  a  number  is  to  be  raised  to  a  power,  a 
small  figure,  called  the  index  or  exponent,  is  placed  above  and 
to  the  right  of  the  number  whose  power  is  to  be  found,  as  42. 
Here  the  exponent  is  2,  and  denotes  the  2d  power  of  4, 
or  4x4.  So33=3x3,  &c. 

The  second  power  of  a  number  1  foot=12  inches, 

is  called  the  square  of  that  num- 
ber, because  the  surface  of  a  geo- 
metrical square  may  be  obtained 
by  multiplying  the  number,  ex- 
pressing one  side,  by  itself.  Thus,  j 
if  the  side  of  the  adjacent  square 
is  12  linear  units,  or,  as  here,  12 
inches  long,  its  entire  surface  will 
be  denoted  by  12  X  12=144  square 
units,  which  in  this  case  will  be  144 
square  inches. 

22 


254  INVOLUTION  AND  EVOLUTION.       [CHAP.  XVI. 

Tlie  third  power  of  a  number  is  3  feet- 

called  the  cube  of  that  number,  be- 
cause the  solid  contents  of  a  geomet- 
rical cube,  as  hi  the  adjacent  figure, 
can  be  obtained  by  raising  the  num- 
ber expressing  one  side,  to  the  3d 
power.  Thus,  3X3X3=27  cubic 
feat. 

To  raise  a  number  to  any  power,  multiply  the  number  by 
itself  as  many  times  as  there  are  units  less  one  in  the  expo- 
nent; the  last  product  will  be  the  power  sought. 

EXAMPLES. 

1-10.  Find  the  square  of  each  of  the  following  numbers  : 
14;  19;  24;  36;  48;  57;  93;  111;  168;  233. 

11-20.  Cube  each  of  the  following  numbers  :  13;  18; 
23;  35;  44;  56;  91;  148;  336;  221. 

21-22.  What  is  the  5th  power  of  47  ?  the  9th  power 
of  9? 

23-26.  What  is  the  square  of  0'75  ?  of  1'14  ?  of  34'09  ? 
of  4-781? 

27-31.  What  is  the  cube  of  0'61  ?  of  0'13  ?  of  0*202  ? 
of  0-65?  of  3-021? 

32-36.  Find  the  the  square  of  2i  ;  of  3|  ;  of  4|  ;  of  7|  ; 


37-41.  Find  the  cube  of  1  ;  of  J  ;  3T4T  ;  of  5-f  ;  of  9f  . 
42-44.  What  is  the  5th  power  of  2-f-  ?  the  4th  power  of 
0-25?  of  0-375? 

EVOLUTION. 

§  135.  Evolution  is  the  reverse  of  involution.  It  is  the 
process  of  finding  the  root  of  a  given  power.  Thus,  6  is  the 
square  'root  of  36,  because  6  raised  to  the  2d  power,  that 


§  136.]  EVOLUTION.  255 

is,  62=36,  is  the  square  of  6.  So  4  is  the  cube  root  of  64, 
because  4  raised  to  the  3d  power,  that  is,  43=64,  is  the  cube 
root  of  4. 

The  symbol  V,  called  the  radical  sign,  is  used  to  denote 
the  square  root  of  a  number,  as  \/Q  —  3;  **/36  =  6.  The 
3d  or  cube  root  is  denoted  by  the  figure  3  written  over  the 
radical  sign,  \S8  =  2  ;  v  64  =  4.  In  like  manner  \/  signi- 
fies the  4th  root,  &c. 

§  136.  Before  explaining  the  method  of  extracting  the  SQOARE  ROOT, 
we  will  involve  some  numbers  by  considering  them  decomposed  into 
units,  tens,  hundreds,  &c. 
What  is  the  square  of  25  ? 

25=20  +5     The  2  tens  are  written  as  20  units. 
20+5 

202+20X5=product  by  the  units  in  the  tens. 
+20X5+52=product  by  the  units. 

202+2  X  20~X  5+52=square  of  20+5. 

That  is,  the  square  of  a  number  consisting  of  tens  and  units  equals 
the  square  of  the  tens  (expressed  in  units),  plus  twice  the  product  of 
the  tens  (expressed  in  units)  into  the  units,  plus  the  square  of  the  units. 

What  is  the  square  of  252  ? 

252=200  +50+2        The  2  hundreds  are  written  as  200  units,  and 
200  +50+2  the  5  tens  as  50  units. 

2002+200X50+200X2=product  by  hundreds  (expressed 

in  units). 

200X50+  503       +  50  X2=product  by  tens  (express- 
ed in  units). 
+200  X2+50X2+22=product  by  units. 

2003+2  X  200  X  50+502+2  X  (200+50)  X  2+2' 

That  is,  the  square  of  a  number  consisting  of  hundreds,  tens,  and 

units,  is  equal  to  the  square  of  the  hundreds  (expressed  in  units),  plus 

twice  tlie  product  of  the  hundreds  into   the  tens  (expressed  in  units), 

plus  the  square  of  the  tens  (expressed  in  units),  plus  twice  the  product 


256  INVOLUTION  AND  EVOLUTION.        [CHAP.  XVI. 

of  the  sum  of  the  hundreds  and  tens  (expressed  in  units')  into  the  units, 
plus  the  square  of  the  units. 

Continuing  in  this  way,  we  could  show  that  the  square  of  the  sum 
of  any  number  of  numbers  is  the  square  of  the  first  number,  plus  twice 
the  product  of  the  first  number  into  the  second,  plus  the  square  of  the 
second  ;  plus  twice  the  product  of  the  sum  of  the  first  two  into  the  third, 
plus  the  square  of  the  third  ;  plus  twice  the  product  of  the  sum  of  the 
first  three  into  the  fourth  ;  plus  the  square  of  the  fourth  ;  and  so  on. 

§  137.  We  will  now  extract  the  square  root  of  625.  But  first  let 
us  ascertain  how  many  figures  its  root  must  have. 

The  smallest  digit  is  1  ;  its  square  is  1.  The  largest  digit  is  9 ;  its 
square  is  81.  The  square  of  the  units,  then,  must  be  either  one  or 
two  figures  ;  either  units  or  units  and  tens. 

The  smallest  number  of  2  figures  is  10 ;  its  square  is  100.  The 
largest  number  of  2  figures  is  99  ;  its  square  is  9801.  The  square  of 
2  figures  or  tens  must,  then,  be  3  figures  or  hundreds,  or  4  figures  01 
thousands,  <fec. 

Hence,  if  a  number  consist  of  1  or  2  figures,  its  root  must  consist  of 
1  figure  ;  if  of  3  or  4  figures,  its  root  must  consist  of  2  figures  ;  if  of 
6  or  6  figures,  its  root  must  consist  of  3  figures,  and  so  on.  That  is, 
the  square  will  contain  twice  as  many  figures  as  the  root,  or  twice  as 
many,  less  one. 

Then,  dividing  any  number  into  groups  by  placing  a  dot  over  the 
first  or  unit  figure,  and  one  over  each  second  figure  towards  the  left, 
we  shall  have  as  many  figures  in  the  root  as  there  are  dots.  Thus 
the  square  root  of  7620016  will  contain  4  figures;  of  625,  2  figures, 
units,  and  tens. 

Then  625=square  of  the  tens  plus  Tens, 

twice  the  tens  into  the  units  plus  the 


square  of  the  units.  The  square  of 
the  tens  must  be  found  in  the  hun- 
dreds of  the  number.  The  greatest 
number  of  tens  whose  square  can  be 
found  in  6  (hundreds)  is  2  tens, 
which  we  write  in  the  quotient.  Subtracting  the  square  4  (hundreds) 
our  remainder  is  225=twice  the  tens  into  the  units  plus  the  square 
of  the  units,  or,  what  is  the  same  thing  (twice  the  tens  plus  the  units), 
Xthe  units  To  find  the  units'  figure,  by  way  of  trial,  we  divide  225 


2  625(25 


Twice  the 

tens.        Units. 
4+5          4 

=45          225 
225 


§  138.] 


EVOLUTION. 


257 


M                                       P 

(400+80;  X7              72 

400X80         H 

K 

802 

c 

G~ 

1   1 

4002 

$    X 

0       ~J 

E    L, 


by  twice  the  tens  ;  the  quotient  is  5.  Adding,  then,  these  units,  5,  to 
twice  the  tens  or  40,  we  have  45  ;  which,  multiplied  by  5,  will  give 
a  product  that  exactly  equals  225. 

Before  giving  a  rule  for  the  extraction  of  the  square  root,  it  may  be 
well  to  illustrate,  geometrically,  the  involution  and  evolution  of  a  num- 
ber. 

4872=(400+80+7)2=4002-{-2X40CXSOX802  +  2  X(400-f80) 
X7  +  72. 

The  square  ABCD  may  be  en- 
larged to  the  square  AEKF,  by  the 
addition  of  the  two  equal  rectangles 
BG  and  DH,  whose  lengths  axe  each 
equal  ^o  the  side  AB  of  the  original 
square,  and  whose  widths  are  equal 
to  BE,  the  quantity  by  which  the  side 
of  the  square  has  been  augmented, 
also  a  little  square,  CGKH,  whose 
side  is  the  same  as  the  width  of  one 
of  the  equal  rectangles. 

Again,  the  square  AEKF,  having  its  side  increased  by  EL,  be- 
comes augmented  by  the  two  rectangles  EN",  FP,  and  the  little  square 
KR.  Thus  we  might  continue  to  augment  the  square  last  found  by 
the  addition  of  two  equal  rectangles,  and  a  little  square  ;  the  length 
of  each  rectangle  being  equal  to  the  side  of  the  square  which  is  to  be 
augmented,  and  the  width  equal  to  the  quantity  by  which  the  side 
of  the  square  is  increased  ;  and  the  side  of  the  little  square  being  the 
same  as  the  width  of  one  of  the  rectangles.  The  diagram  is  adapted 
to  the  case  of  squaring  4004-804-7=487. 

§  138.  We  will  now  reverse  the  above  process. 

Let  it  be  required  to  extract  the  square  root  of  527076  ;  that  is, 
we  will  seek  the  number  of  feet  in  the  side  of  a  square  whose  area 
shall  contain  527076  square  feet. 

In  this  example  there  must  be  three  figures  in  the  root. 

We  know  that  the  side  of  the  square  sought  must  exceed  700  linear 
feet,  since  the  square  of  700  is  490000,  which  is  less  than  527076  ;  we 
also  know  that  the  side  of  this  square  must  be  less  than  800  linear  feet, 
since  the  square  of  800  is  640000,  which  is  greater  than  527076.  Hence 
the  first,  or  hundreds'  figure  of  the  root,  is  7 ;  which  is  the  greatest  num 
ber  whose  square  can  be  contained  in  52,  the  first  or  left-hand  period. 

22* 


258 


INVOLUTION  AND  EVOLUTION.        [CHAP.  XVI. 


F  !  - 


If  we  suppose  each 
side  of  the  square 
ABCD  to  be  700  lin- 
ear feet,  its  surlace 
will  be  700X700= 
490000  square  feet, 
which,  subtracted 
from  627076  square 
feet,  leaves  37076 
square  feet. 

Hence  it  is  neces- 
sary to  increase  the 
square  ABCD,  by 
37076  square  feet. 
We  have  seen  that 
such  increase  is  effect- 
ed by  the  addition  of  EL 
two  equal  rectangles,  and  a  little  square.  The  surface  of  the  two 
rectangles  will  evidently  make  by  far  the  largest  portion  of  the  whole 
increase.  The  length  of  one  of  these  rectangles  is  the  same  as  the 
length  of  a  side  of  the  square  ABCD,  which  has  already  been  shown 
to  be  700  linear  feet.  The  length  of  the  two  rectangles  taken  to- 
gether will  be  twice  700  linear  feet,  or,  what  would  be  the  same 
thing,  700  linear  feet  added  to  700  linear  feet.  If  to  BC,  which  is 
700  linear  feet,  we  add  CD,  which  is  also  700  linear  feet,  we  shall 
have  BC-f-CD,  equal  to  1400  linear  feet,  for  the  length  of  the  two 
rectangles.  Were  we  to  multiply  1400  by  the  width  of  a  rectangle, 
we  should  obtain  the  number  of  square  feet  in  these  rectangles,  or 
nearly  the  37076  square  feet  which  require  to  be  added.  Conse- 
quently, if  we  divide  37076  by  1400,  the  quotient  will  give  the  ap- 
proximate width  of  the  rectangles.  Using  1400  as  a  trial  divisor,  we 
find  it  to  be  contained  between  20  and  30  times  in  37076  ;  hence  the 
second  or  ten's  figure  of  the  root  is  2.  But  besides  the  rectangles, 
there  must  be  added  the  little  square  CGKH,  each  side  of  which  is 
20  linear  feet ;  we  will  therefore  add  20  to  1400,  and  thus  obtain 
1420  for  the  total  length  of  the  two  rectangles  and  the  side  of  the 
little  square.  Now,  multiplying  1420  by  20,  we  obtain  28400  square 
feet  for  the  total  additions,  which,  subtracted  from  37076,  leaves 


§  138.]  EVOLUTION.  259 

8676  square  feet.  The  square  AEKF  thus  completed  is  720  feet  on 
a  side. 

Again,  a  side  of  this  square  is  to  be  further  increased  so  that  the 
added  surface  will  amount  to  8676  square  feet.  And,  as  before,  the 
parts  added  will  consist  of  two  equal  rectangles  and  a  little  square. 
The  trial  divisor,  which  is  the  sum  of  the  length  of  the  two  new  rect- 
angles, is  the  same  as  the  sum  of  two  sides  of  the  square  AEKF. 

If,  now,  to  1420  already  found,  we  add  20,  we  shall  have  1440, 
which  is  the  sum  of  EK  and  KF,  and  which  is  our  second  trial  divi- 
sor. We  find  this  divisor  contained  between  6  and  7  times  in  8676  ; 
hence  our  third  or  units'  figure  of  the  root  is  6.  Therefore  6  is  the 
width  of  the  second  set  of  rectangles.  The  second  little  square 
KNRP,  of  the  same  width  as  the  rectangles,  must  be  6  linear  feet  on 
a  side  ;  therefore,  adding  6  to  1440,  we  find  1446  for  the  whole  length 
of  the  new  rectangles  and  a  side  of  the  second  little  square.  Multi- 
plying 1446  by  6,  we  obtain  8676  square  feet  as  the  sum  of  the  se- 
ries of  additions  to  the  square  AEKF,  thus  forming  the  square  ALRM, 
which  is  the  square  sought;  each  side  being  726  feet. 

The  above  work  may  be  arranged  as  follows : 

NUMBER.  ROOT. 

Linear  feet.  Square  feet.        Linear  feet. 

700  527076(700+20+6=726 

1400=lst  trial  divisor       490000 


1420  37076 

1440=2d  trial  divisor         28400 

1446  8676 

8676 

0 

If  we  omit  the  ciphers  on  the  right,  the  work  will  take  the  follow- 
ing condensed  form : 

NUMBER.     ROOT. 

Linear  feet.  Square  feet.     Linear  feet. 

7  527076  (726 

14  =lst  trial  divisor  49 

142  ^370 

144=2d  trial  divisor  284 

1446  8676 

8676 

0 


260  INVOLUTION  AND  EVOLUTION.        [CIIAP.  XVI. 

CASE  I. 

From  the  above  process,  we  deduce  the  following  rule  for 
the  extraction  of  the  square  root  of  a  whole  number. 

RULE. 

I.  Separate  the  given  number  into  periods  of  two  figures 
each,  counting  from  the  right  towards   the  left.      When  the 
number  of  figures  is  odd,  it  is  evident  that  the  left-hand,  or 
first  period,  will  consist  of  but  one  figure. 

II.  Find  the  greatest  square  in  the  first  period,  and  place 
its  root  at  the  right  of  the  number,  in  the  form  of  a  quotient 
in  division,  also  place  it  at  the  left  of  the  number.     Subtract 
the  square  of  this  root  from  the  first  period,  and  to  the  re- 
mainder annex  the  second  period  •  the  result  will  be  the  FIRST 
DIVIDEND. 

III.  To  the  figure  of  the  root,  as  placed  at  the  left  of  the 
number,  add  the  figure  itself,  and  the  sum  will  be  the  FIRST 
TRIAL  DIVISOR.     See  how  many  times  this  trial  divisor,  with 
a  naught  annexed,  is  contained  in  the  dividend  ;  the  quotient 
will  be  the  next  figure  of  the  root ;  this  must  be  annexed  to 
the   TRIAL  DIVISOR;    the   result  will  be  the  TRUE  DIVISOR. 
Multiply  the  true  divisor  by  this  last  figure  of  the  root,  and 
subtract  the  product  from  the  dividend,  and  to  the  remainder 
annex  the  next  period,  for  a  NEW  DIVIDEND. 

IV.  To  the  last  TRUE  DIVISOR  add  the  last  figure  of  the 
root  j  the  sum  will  be  a  new  TRIAL  DIVISOR.      Continue  to  op- 
erate as  before,  until  all  the  periods  have  been  brought  down. 

NOTE  1. — In  case  any  dividend  is  not  so  great  as  its  trial  divisor, 
with  a  cipher  annexed,  write  0  as  the  next  figure  of  the  root ;  also 
place  0  at  the  right  of  the  divisor,  and  form  a  new  dividend  by  an- 
nexing a  new  period. 

NOTE  2. — Approximate  roots  may  be  found  by  annexing  decimal 
periods  of  two  naughts  each. 


§  138.]  EVOLUTION.  261 

What  is  the  square  root  of  11390625? 

3  li390625(3375 

63  9 

667 

6745 


EXAMPLES. 


45.  What  is  the  square  root  of  11019960576  ? 

46.  What  is  the  square  root  of  276793836544  ? 
47-52.  What  is  the  square  root  of  12321?  of  53824? 

of  30858025  ?  of   16983563041 ?    of  852891037441 ?  of 
61917364224? 

CASE    II. 

To  extract  the  square  root  of  decimals. 

Annex  one  cipher,  if  necessary,  so  that  the  number  of  deci- 
mals shall  be  even  ;  then,  point  them  off  into  periods  of  two 
figures  each,  counting  from  the  units'  place  towards  the  right. 
Extract  the  root,  as  in  Case  I. 

NOTE. — If  the  given  number  has  not  an  exact  root,  there  will  be  a 
remainder  after  all  the  periods  have  been  brought  down,  in  which 
case  the  operation  may  be  extended  by  forming  new  periods  of 
ciphers. 

EXAMPLES. 

53.  What  is  the  square  root  of  3486-78401  ? 


262  INVOLUTION  AND  EVOLUTION.         [CHAP.  XVI. 

54-57.  What  is  the  square  root  of  6 '553 6  ?  of 
0-00390625?  of  17?  of  37'5  ? 

58.  What  is  the  square  root  of  0'0000012321  9 

CASE  III. 

To  extract  the  square  root  of  a  common  fraction,  or  mixed 
number. 

Reduce  the  vulgar  fraction,  or  mixed  number,  to  its  sim- 
plest fractional  form.  Extract  the  square  root  of  the  numer- 
ator and  denominator  separately,  if  they  have  exact  roots  ;  if 
they  have  not,  reduce  the  fraction  to  a  decimal,  and  proceed 
as  in  Case  II. 

EXAMPLES. 

59-60.  What  is  the  square  root  of  ff  J?  of  }^\  ? 
61-62.  What  is  the  square  root  of  f  of  £  of  f  of  J  ?  of 
4J1? 

63-65.    What  is  the  square  root  of  4f?  of  \\  ?  of  if  ? 

EXAMPLES  INVOLVING    THE    PRINCIPLES  OF  THE  SQUARE  ROOT. 

§  139.  A  triangle  is  a  figure  having  three  sides,  and  con- 
sequently three  angles. 

When  one  of  the  angles  is  right,  like  the  corner  of  a 
square,  the  -triangle  is  called  a  right-angled  triangle.  In 
this  case  the  side  opposite  the  right  angle  is  called  the  hy- 
potenuse. 

It  is  an  established  proposition  of  geometry,  that  the  square 
of  the  hypotenuse  is  equal  to  the  sum  of  the  squares  of  the 
other  two  sides. 

Hence  itfoUoivs  that  the  square  of  the  hypotenuse,  dimin- 
ished by  the  square  of  one  of  the  sides,  equals  the  square  of 
the  other  side. 


§  139.]  KYOLUTION.  263 

By  means  of  these  properties,  it  follows  that  two  sides  of 
t\  right-angled  triangle  being  given,  the  third  side  can  be 
found. 

How  long  must  a  ladder  be  to  reach  to  the  top  of  a  house  40  feet 
high,  when  the  foot  of  it  is  30  feet  from  the  house  ? 

In  this  example,  it  is  obvious  that  the  ladder  forms  the  hypotenuse 
of  a  right-angled  triangle,  whose  sides  are  30  and  40  feet  respective- 
ly.   Therefore,  the  square  of  the  length  of  the  ladder  must  equal  the 
sum  of  the  squares  of  30  and  40. 
302=  900 
40^=1600 

\/2500=50,  the  length  of  the  ladder. 

EXAMPLES. 

G6.  Suppose  a  ladder  100  feet  long,  to  be  placed  60  feet 
from  the  roots  of  a  tree  ;  how  far  up  the  tree  will  the  top 
of  the  ladder  reach  ? 

67.  Two  persons  start  from  the  same  place,  and  go,  the 
one  due  north,  50  miles,. the  other  .due  west,  80  miles.  How 
far  apart  are  they  ? 

68.  What  is  the  distance  through  the  opposite  corners  of 
a  square  yard  ? 

.  69.  The  distance  between  the  lower  ends  of  two  equal 
rafters,  in  the  different  sides  of  a  roof,  is  32  feet,  and  the 
height  of  the  ridge  above  the  foot  of  the  rafters  is  12  feet. 
What  is  the  length  of  a  rafter  ? 

70.  What  is  the  distance  measured  through  the  centre  of 
a  cube,  from  one  corner  to  its  opposite  corner,  the  cube  be- 
ing 3  feet,  or  one  yard,  on  a  side  ? 

We  know,  from  the  principles  of  geometry,  that  all  similar 
surfaces,  or  areas,  are  to  each  other  as  the  squares  of  their 
like  dimensions. 

7 1 .  Suppose  we  have  two  circular  pieces  of  land,  the  one 


264  INVOLUTION  AND  EVOLUTION.       [CHAP.  XVI. 

100  feet  in  diameter,  the  other  20  feet  in  diameter.     How 
much  more  land  is  there  in  the  larger  than  in  the  smaller  ? 
NOTE. — The  circles  will  be  to  each  other  as  the  squares  of  their 
respective  diameters. 

72.  Suppose,  by  observation,  it  is  found  that  4  gallons  of 
water  flow  through  a  circular  orifice  of  1  inch  in  diameter 
in  1  minute.     How  many  gallons  would,  under  similar  cir- 
cumstances, be  discharged  through  an  orifice  of  3  inches  in 
diameter,  in  the  same  length  of  time  ? 

73.  What  length  of  thread  is  required  to  wind  spirally 
around  a  cylinder,  2  feet  in  circumference  and  3  feet  in 
length,  so  as  to  go  but  once  around  ? 

NOTE. — It  is  evident  that  if  the  cylinder  be  placed  upon  a  plane, 
and  be  caused  to  roll  once  over,  it  will  describe  a  rectangle,  whose 
width  is  2  feet,  and  length  3  feet ;  while  the  thread  will  form  its 
diagonal,  or  line  running  from  corner  to  corner. 

74-77.  A  room  is  16  feet  long,  12  feet  wide,  and  10  feet 
high.  What  is  the  diagonal  distance  measured  on  the  floor  ? 
What  is  the  diagonal  distance  measured  on  the  wall  which 
forms  the  side  of  the  room  ?  What  is  the  diagonal  distance 
measured  on  the  wall  which  forms  the  end  of  the  room  ? 
And  what  the  diagonal  passing  through  the  centre  of  the 
room? 

78-80.  There  are  three  circular  pieces  of  ground,  the 
diameters  of  which  are  697  feet,  185  feet,  153  feet.  What 
is  the  diameter  of  a  circular  piece  whose  area  is  equal  to  the 
difference  between  the  first  and  second  ?  What  the  diame- 
ter of  one  whose  area  is  equal  to  the  difference  between  the 
first  and  third  ?  What  the  diameter  of  one  whose  area  is 
equal  to  the  difference  between  the  second  and  third  ? 

§  140.  Before  explaining  the  method  of  extracting  the  CUBE  ROOT, 
we  will  involve  the  number  45,  consisting  of  4  tens  and  5  units  to  the 
third  power. 


§  141.]  EVOLUTION.  265 

45=40  -f  5 
40+5 


402+40X5=product  by  the  units  in  the  tens. 
40X5+52=product  by  the  units. 

402+2X40X5+52=squaro  of  40 -f 5 
40  +5 

403+2X402X5+40X52 

402X5+  2X40X52+53 

403+3X402X5+3X40X52+53=cube  of  40+5. 
That  is,  the  cube  of  a  number  consisting  of  tens  and  units,  equals 
the  cube  of  the  tens  (expressed  in  units),  plus  three  times  the  square  of 
the  tens  (expressed  in  units')  into  the  units,  plus  three  times  the  units 
in  the  tens  into  the  square  of  the  units,  phis  the  cube  of  the  units. 
And  in  general,  the  cube  of  the  sum  of  any  number  of  numbers  is 
equal  to  the  cube  of  the  first  number,  plus  three  times  the  square  of 
the  first  number  into  the  second,  plus  three  times  the  first  into  the 
square  of  the  second,  plus  the  cube  of  the  second ;  plus  three  times  the 
square  of  the  SUM  OF  THE  FIRST  TWO  into  tJie  third,  plus  three  times  the 
SUM  OF  THE  FIRST  TWO  into  the  square  of  the  third,  plus  the  cube  of  the 
third,  &c. 

EXAMPLES. 

81-90.  Express  by  symbols  as  above,  753 ;  89s ;  1423 ; 
365s;  473;  963 ;  2213;  4963;  8793 ;  9993. 

§  141.  We  will  now  extract  the  cube  root  of  91125.  It 
is  first  to  be  determined  how  many  figures  the  root  must 
have. 

The  smallest  number  of  2  figures  is  10 ;  its  cube  is  1000. 

The  largest  number  of  2  figures  is  99  ;  its  cube  is  970299. 

A  number,  then,  consisting  of  4  to  6  figures  must  have  a  root  of  2 
figures.  So,  a  number  consisting  of  7  to  9  figures,  must  have  a  root 
of  3  figures.  If,  then,  we  divide  a  number  into  groups,  by  placing  a 
dot  over  the  first  or  unit  figure,  and  one  over  each  3d  figure  towards 
the  left,  we  shall  have  as  many  figures  in  the  root  as  there  aro  dots, 
91125  has  therefore  2  figures  in  its  root. 

23 


266 


INVOLUTION  AND  EVOLUTION.          [CHAP.  XVI 


91125(45 
64 


3Xsq.  of  tens  =4800  trial  div.    27125 
3XtensXunits=  600  27125 

sq.  of  units       =     25 

5425  true  divisor. 


The  root  must  con- 
sist of  tens  and  units ; 
then  91125=cube  of 
the  tens  plus  3  times 
the  square  of  the 
tens  into  the  units 
plus  3  times  the  tens 
into  the  square  of 
the  units  plus  the  cube  of  the  units.  The  cube  of  the  tens  will  be 
found  in  the  91  (thousand).  The  greatest  cube  contained  in  91  is  4 
(tens),  which  we  write  in  the  quotient.  Subtracting  the  cube  64 
(thousands),  our  remainder  is  27125=3  times  the  square  of  the  tens 
into  the  units  plus  3  times  the  tens  into  the  square  of  the  units  plus 
the  cube  of  the  units.  Or,  what  is  the  same  thing  (3  X  square  of  the 
tens  plus  3Xthe  tens  X  the  units  plus  the  square  of  the  units),  as  one 
factor  X  the  units  as  the  other  factor. 

To  find  the  units'  figure  of  the  root,  by  way  of  trial,  we  divide 
27125  by  3 X square  of  the  tens;  that  is,  by  4800.  The  quotient  is  5. 

Having  now,  as  we  suppose,  the  correct  units'  figure  of  the  root  as 
the  one  factor,  let  us  ascertain  the  value  of  the  other  factor,  or  the 
true  divisor.  3Xsq.  tens=3  XI  600=4800  ;  3  X  tens  X  the  units=«0> 
X  40X5=600;  sq.  of  units=52=25.  Then  4800+600-f  25=5425, 
the  completed  divisor.  This,  multiplied  by  5,  the  quotient  figure, 
gives  27125  :  a  result  showing  that  our  quotient  figure  was  neither 
too  large  nor  too  small.  The  cube  root,  then,  of  91125  is  45. 

Before  giving  a  rule  for  the  extraction  of  cube  root,  we  will  illus- 
trate geometrically  the  involution  and  the  evolution  of  a  number. 

Let  it  be  required  to  cube  45,  the  number  before  employed,  or 
suppose  we  are  required  to  find  the  number  of  cubic  inches  in  a  cube 

whose    side    is     45 

Fig.  1. 
inches.      Separating 

45  into  40+5,  we 
will  suppose  the 
cube  (fig.  1)  to  be  40 
inches  on  a  side ; 
then  40  X  40  X  40 
will  give  the  solid 
contents  of  this  cube, 
represented  by  40s. 


403=40X  40X40 
=64000 


§141.] 


EVOLUTION. 


267 


24000 


Fig.  3. 


3X40=120 
X  by  52  =  25 

600 
249 


Let  fig.  2  represent  Fig.  2. 

the  cube  increased 
by  three  equal  slabs; 
then  3  (the  number 
of  slabs)  times  40a 
(the  surface  of  one  of 
the  slabs)  multiplied 
by  5,  the  thickness  of 
a  slab,  will  give  the 
solid  contents  of  the 
slabs,  represented  by 
3X402X5. 

Let  fig.  3  represent 
the  solid  (as  in  fig. 
2),  further  increased 
by  three  equal  corner 
pieces ;  then  3  (the 
number  of  corner 
pieces)  times  40  (the 
length  of  one  corner 
piece)  multiplied  in- 
to 52,  the  surface  of 
an  end  of  a  corner 
piece,  will  give  the 
solid  contents  of  the 
corner  pieces  repre- 
sented by  3X40X 
52. 

Let  fig.  4  represent 
the  solid  (as  in  fig. 
3),  further  increased 
by  a  little  corner 
cube,  each  side  of 
which  is  5  inches ;  J;hen  5X5X5  will  give  the  solid  contents  of  this 
cube,  represented  by  53. 

Then  the  whole  cube  thus  increased  will  be  represented  by  453= 
403+3X4l/lX5+3X40X52+53 

=64000+24000+3000+125=91125. 


3000 


Fig.  4. 


268 


INVOLUTION  AND  EVOLUTION.        [CHAP.  XVI. 


Fig.  1. 


§  142.  Let  it  now  be  required  to  find  the  cube  root  of  382657176 
"We  will  suppose  382657176  to  denote  the  number  of  cubic  feet  in  a 
geometrical  cube  ;  we  are  required  to  find  the  number  of  linear  feet 
in  a  side  of  this  cube,  that  is,  the  length  of  one  of  its  sides.  382657176 
must  give  3  figures  for  the  root. 

We  know  that  the  side  of  the  cube  sought  must  exceed  700  linear 
feet,  since  the  cube  of  700  is  343000000,  which  is  less  than  382657176  ; 
we  also  know  that  the  side  of  this  cube  must  be  less  than  800  linear 
feet,  since  the  cube  of  800  is  512000000,  which  is  greater  than 
382657176.  Hence  the  first  figure  of  our  root,  or  the  figure  in  the 
hundreds'  place,  is  7  ;  whose  cube,  343,  is  the  greatest  cube  contained 
in  382,  the  first  or  left-hand  period. 
If  we  suppose  each  side  of  the  cube, 
represented  by  figure  1,  to  be  700 
linear  feet,  one  of  the  equal  faces, 
as  the  upper  face  DEFG,  will  be 
denoted  by  700  X  700  =  490000 
square  feet.  The  solid  contents  of 
the  cube  will  be  represented  by  700 
X  700  X  700  =  7002  X  700  =  490000 
X  700—343000000  cubic  feet.  Sub- 
tracting 343000000  cubic  feet  from  382657176  cubic  feet,  we  find 
39657176  cubic  feet  for  a  remainder. 

Hence  it  is  necessary  to  increase  the  cube  (figure  1),  by  39657176 
cubic  feet.  We  have  seen  that  such  increase  is  effected  by  the  ad- 
dition of  three  equal  slabs,  three  equal  corner  pieces,  and  an  addition- 
al cube ;  and  that  the  contents  of  the  three  slabs  will  make  by  far  the 
largest  portion  of  the  whole  increase. 

The  number  of  square  feet  in  the 
face  of  one  of  these  slabs  will  be 
the  sain  j  as  the  'number  of  square 
feet  in  the  face  of  the  cube  (figure 
1),  which  has  already  been  shown 
to  be  490000  square  feet.  The  sur- 
face of  the  three  slabs  will  be  three 
times  490000  square  feet ;  or,  which 
would  be  the  same  thing,  twice 
490000  square  feet,  added  to  490000 


Fig.  2. 


§  142.]  EVOLUTION.  269 

square  feet.*  If  to  AB  (fig.  1),  which  is  700  linear  feet,  we  add 
BC,  which  is  also  700  linear  feet,  we  shall  have  AB+BC  equal  to 
1400  linear  feet,  which,  multiplied  by  DB,  equal  to  700  linear  feet, 
will  give  980000  square  feet,  for  the  area  ABDG+BCED,  which, 
added  to  DEFG,  which  is  490000  square  feet,  will  give  1470000 
square  feet,  for  the  area  of  three  faces  of  the  cube  (figure  1),  which  is 
the  same  as  the  area  of  the  three  slabs.  "Were  we  to  multiply  1470000 
by  the  thickness  of  the  slabs,  we  should  obtain  the  cubic  feet  in  these 
slabs.  And  since  the  contents  of  the  slabs  make  nearly  the  whole 
amount  added,  it  follows  that  1470000  multiplied  by  the  thickness 
of  slabs,  will  give  nearly  39657176  cubic  feet.  Consequently  t  rf  we 
divide  39657176  by  1470000,  the  quotient  will  give  the  approximate 
thickness  of  the  slabs.  Using  1470000  as  a  trial  divisor,  we  find  it 
to  be  contained  between  20  and  30  times  in  39657176;  hence  the 
second  or  tens'  figure  of  the  root  is  2. 

We  have  already  remarked  that  1470000  multiplied  by  20,  the 
thickness  of  the  slabs,  will  give  their  solid  contents.  But  besides  the 
slabs,  there  must  be  added  three  corner  pieces,  each  of  which  is  700 
feet  long,  and  of  the  same  thickness  as  the  slabs,  that  is,  20  feet. 
Since  each  corner  piece  is  the  same  length  as  a  side  of  the  cube,  fig- 
ure 1,  it  follows  that  adding  700  to  1400  or  700+700,  the  sum  2100 
will  represent  the  total  length  of 
the  three  corner  pieces.  Were  we 
to  multiply  2 100  by  20,  we  should 
obtain  the  area  of  the  three  cor- 
ner  pieces,  which  might  be  added 
to  1470000,  the  area  of  the  three 
slabs.  But,  since  there  is  also  to 
be  added  a  little  cube,  each  of 
whose  sides  is  20  linear  feet,  we 
will  add  20  to  2100,  and  thus  ob- 
tain #120  for  the  total  length  of 
the  three  corner  pieces,  and  of  a 


*  It  will  be  noted  that  the  peculiar  steps  throughout  this  demonstration  have 
reference  to  the  mode  of  extracting  the  Cube  Root  which  follows.  The  object  of 
these  processes  is,  to  make  use  of  what  has  been  obtained  in  one  ftage  of  tho 
work  for  the  stage  next  succeeding ;  to  obtain  a  new  quantity  by  adding  to  one  al- 
ready in  hand,  instead  of  m.-:ftiplyiiiff  an  original  quantity;  thereby  saving  much 
time  and  labor. 


270 


INVOLUTION  AND  EVOLUTION         [CHAP.  XVI. 


side  of  the  little  cube.  Now,  multiplying  2120  by  20,  we  obtain 
42400  square  feet  for  the  surface  of  the  three  corner  pieces  and  a 
face  of  the  little  cube;  which,  added  to  1470000,  the  number  of 
square  feet  in  the  faces  of  the  three  slabs,  will  give  1512400  square 
feet  in  all  the  additions.  'If  we  multiply  1512400  by  20,  the  thick- 
ness of  these  additions,  we  shall  obtain  30248000  cubic  feet  for  all 
the  additions,  which,  subtracted  from  39657176,  leaves  9409176  cubic 
feet.  The  cube  thus  completed  is  720  feet  on  a  side,  and  is  repre- 
sented by  figure  4. 

Figure  a. 


Fig.  4. 


The  surfaces  now  obtained  may  be  represented  (figure  a)  by  the 
parts  included  within  the  heavy  lines.  The  three  divisions  of  the 
figure,  including  the  dotted  lines,  may  be  supposed  to  be  three  entire 
faces  of  the  cube,  figure  4. 

But  this  cube  is  to  be  further 
increased  by  9409176  cubic  feet. 
And  as  before,  the  parts  added 
will  consist  of  three  equal  slabs, 
three  equal  corner  pieces,  and  a 
little  cube.  The  trial  divisor,  which 
is  the  area  of  the  faces  of  the  three 
slabs,  is  the  same  as  three  times 
the  area  of  a  face  of  the  cube,  fig- 
ure 4,  each  of  whose  sides  is  720 
feet. 

Now,   to   obtain  this   area,  we 

have  only  to  add  to  the  surfaces  already  obtained,  and  represented 
within  the  heavy  lines  (figure  a),  three  rectangles,  each  700  feet  by 
20,  and  two  little  squares  20  feet  by  20  feet. 

If  to  2120,  a  number  which  we  already  have,  we  add  20,  we  shall 


f  <7L  -f-  , 


§  142.]  EVOLUTION.  271 

obtain  2140,  the  linear  extent  of  the  rectangles  and  squares  desired, 
as  in  the  dotted  portions  (figure  a).  And  as  these  dotted  portions 
have  all  the  same  width  of  20  feet,  if  we  multiply  2140  by  20,  we 
shall  obtain  42800  square  feet  for  the  area  of  the  dotted  portion 
(figure  a),  which,  added  to  1512400,  the  area  of  the  parts  included 
within  the  heavy  lines,  will  give  1555200  square  feet  for  the  area  of 
three  slabs,  each  equal  to  one*  face  of  the  cube  (figure  4).,  This  will 
be  a  second  trial  divisor.  We  find  this  divisor  contained  between  6 
and  7  times  in  9409176  ;  hence  our  third  figure  of  the  root,  or  the 
figure  in  the  units'  place,  is  6.  "Were  we  to  multiply  1555200  by 
6,  it  would  give  the  cubic  feet  in  the  second  set  of  slabs.  But  before 
multiplying,  we  will  increase  that  sum  by  the  surface  of  the  second 
set  of  corner  pieces,  and  of  the  second  little  cube.  The  length  of 
each  corner  piece  is  the  same  as  a  side  of  the  cube,  figure  4,  which  is 
720  feet;  hence,  adding  20  to  2140  already  found,  we  obtain  2160, 
which,  being  3  times  720,  will  be  the  linear  extent  of  the  three  cor- 
ner pieces.  Were  we  to  multiply  2160  by  6,  we  should  find  the  sur- 
face of  these  three  corner  pieces,  but  as  we  wish  also  the  area  of  one 
of  the  faces  of  the  second  little  cube,  we  add  6  to  2160,  and  thus  ob- 
tain 2166,  which,  multiplied  by  6,  will  give  12996  for  surface  of  sec- 
ond set  of  corner  pieces  and  of  second  little  cube  ;  this  added  to 
1555200,  gives  1568196  for  the  surface  of  the  whole  second  series  of 
additions.  Multiplying  1568196  by  6,  we  obtain  9409176  cubic  feet, 
which  have  thus  been  added  to  the  cube  represented  by  figure  4  ; 
hence  the  cube  whose  side  is  726  feet  is  the  one  sought.  The  above 
work  may  be  arranged  as  follows  : 


1ST  COLUMN.  2D  COLUMN.  NUMBER.                     ROOT. 

linear  feet.  Square  feet.  Cubic  feet.            Linear  feet. 

700  490000  382657176(700+20+6=726 

1400  1470000=lst  tr.  divisor,  343000000 

2100  1512400                               39657176 

2120  1555200=2d  tr.  divisor,     30248000 


2140  1568196  9409176 

2160  9409176 

2166  0~ 


272  INVOLUTION  AND  EVOLUTION.        [CHAP.  XVI. 

If  -we  omit  the  ciphers  on  the  right,  and  omit  unnecessary  terms, 
the  work  will  take  the  following  condensed  form : 

1ST  CQLUMN.  2D  COLUMN.                                            NUMBER.      ROOT. 

Linear  feet.  Square  feet.  Cubic  feet.    Linear  feet. 

7*'  49  382657176(  726 

14  147=1  st  trial  divisor,  346      • 

212  15124  39657 

214  15552=2d  trial  divisor,  30248 


2166       1568196  9409176 

9409176 


NOTE. — In  the  extraction  of  the  cube  root,  as  just  illustrated,  it  will 
be  noticed  that  each  divisor  is  a  geometrical  surface ;  that  is  to  say, 
the  product  of  two  dimensions,  width  and  breadth,  for  example ;  and 
of  course  the  quotient  must  be  the  other  dimension,  that  is,  the  thick- 


But  it  is  important  to  remember  that  it  is  only  squares  and  cubes, 
square  roots  and  cube  roots,  that  can  have  any  relation  to  geometrical 
dimensions  ;  any  higher  power  of  a  number,  as  45,  or  any  other  root 
as  ty,  cannot  be  illustrated  by  blocks.  The  principle,  therefore,  of 
involution  and  evolution  is,  strictly  speaking,  independent  of  surfaces 
and  solids  ;  it  is  purely  arithmetical. 

From  the  foregoing  demonstration  we  may  deduce  the  following 

RULE. 

I.  Separate  the  number  whose  root  is  to  be  found,  into 
periods  of  three  figures  each,  counting  from  the  units'  place 
towards  the  left.    When  the  number  of  figures  is  not  divisible 
by  3,  the  left-hand  period  will  contain  less  than  3  figures. 

II.  Seek  the  greatest  figure  whose  cube  shall  not  exceed  the 
first  or  left-hand  period  ;  write  it  after  the  manner  of  a  quo- 
tient in  division  for  the  first  figure  of  the  root.     Place  this 
figure  for  the  head  of  a  first  left-hand  column,  and  its  square 
for  the  head  of  a  second  left-hand  column,  and  subtract  its 


§  142]  EVOLUTION.  273 

cube  from  the  first  period.  To  the  remainder  bring  down  a 
second  period  for  the  FIRST  DIVIDEND.  Add  the  figure  in  the 
root  to  the  term  of  the  IST  COLUMN  already  found,  for  its 
next  term,  which  multiply  by  the  same  figure,  and  add  the 
product  to  the  term  already  found  in  the  2o  COLUMN,  for  its 
next  term,  which  will  be  a  TRIAL  DIVISOR. 

(III.  Find  how  many  times  the  trial  divisor,  with  two 
naughts  annexed,  is  contained  in  the  dividend  ;  write  the  quo- 
tient for  the  next  figure  of  the  root.  Annex  this  figure  to  the 
last  term  of  the  IST  COLUMN,  after  having  added  to  that  term 
the  preceding  quotient  figure  ;  this  ivill  give  the  next  term  of 
the  IST  COLUMN.  Multiply  this  term  by  the  last  found  fig- 
ure in  the  root,  and  add  the  product,  after  advancing  it  two 
places  to  the  right,  to  the  last  term  of  the  2o  COLUMN,  for  its 
next  term.  Multiply  this  term  by  the  last  found  figure  of 
the  root,  and  subtract  the  product  from  the  dividend,  and  to 
the  remainder  bring  down  the  next  period  for  a  NEW  DIVI- 
DEND. 

Proceed  as  before  until  all  the  periods  have  been  brought 
down. 

NOTE  1. — When  any  dividend  is  not  so  great  as  the  corresponding 
trial  divisor  with  two  ciphers  annexed,  write  0  for  the  next  figure  of 
the  root,  and  to  the  dividend  bring  down  the  next  period.  Use  the 
same  trial  divisor  as  before,  but  with/owr  ciphers  annexed. 

NOTE  2. — The  trial  divisor,  being  less  than  the  true  divisor,  will 
sometimes  give  too  large  a  quotient  figure  ;  when  the  multiplication 
of  the  true  divisor  by  this  figure  shows  such  to  be  the  case,  this  figure 
must  be  made  smaller. 

NOTE  3. — By  the  above  rule,  which  is  different  from  the  rule  usu- 
ally given  by  the  aid  of  geometrical  diagrams,  we  keep  distinct  all 
the  geometrical  magnitudes ;  thus  our  first  column  represents  the  nu- 
merical values  of  lines,  the  second  column  represents  the  numerical 
values  of  surfaces,  and  the  third  column  corresponds  to  solids.  And, 
;is  vre  are  never  required  to  multiply  by  any  number  greater  than  a 


274 


INVOLUTION  AND  EVOLUTION.        [(JIIAP.  XVI. 


digit,  the  labor  of  multiplying  and  adding  results  to  the  terms  of  the 
successive  columns  is  far  simpler  than  at  first  might  be  supposed. 

By  means  of  these  auxiliary  columns,  the  -work  bears  a  close  analo- 
gy to  Horner's  method  of  solving  numerical  cubic  equations.  (See 
Treatise  on  Algebra.)  The  use  of  auxiliary  columns  becomes  very 
apparent  in  the  extraction  of  roots  of  the  higher  orders,  as  the  fifth 
root,  the  seventh  root,  <kc.  (See  Higher  Arithmetic.) 

What  is  the  cube  root  of  913517247483640899  ? 

1ST  COLUMX.  2D  COLUMN.  NUMBER.  ROOT 


81 


913517247483640899(970299 
729 


18 

277 

284 
29102 

29104 
291069 

291078 
2910879 

243         184517 
26239       183673 

28227 
282328204 

282386412 
28241260821 

28243880523 
2824414250211 

844247483 
564656408 

279591075640 
254171347389 

25419728251899 
25419728251899 

EXAMPLES. 

91.  What  is  the  cube  root  of  10077696  ? 
92-95.    What   is   the    cube  root  of   2357947691  ?    of 
42875?  of  117649?  of  7256313856  ? 

CASE  II. 

To  extract  the  cube  root  of  a  decimal,  annex  ciphers,  if 
necessary,  so  that  the  decimals  may  be  separated  into  equal 
periods  of  3  figures  each. 

Point  off  into  periods  of  3  figures  each,  counting  from 
units  towards  the  right,  and  proceed  as  in  whole  numbers. 


§  143.]  EVOLUTION.  275 

NOTE. — If  the  given  number  has  not  an  exact  root,  there  will  be  a 
remainder  after  all  the  periods  have  been  brought  down.  The  pro- 
cess may  be  continued  by  annexing  ciphers  for  new  periods. 

EXAMPLES. 

96.  What  is  the  cube  root  of  Q'469640998917  ? 

97.  What  is  the  cube  root  of  18'609625  ? 

98.  What  is  the  cube  root  of  1-25992105  ? 

99.  What  is  the  cube  root  of  2  ? 

100.  What  is  the  cube  root  of  9  ? 

101.  What  is  the  cube  root  of  3  ? 

CASE   III. 

To  extract  the  cube  root  of  a  common  fraction,  or  mixed 
number,  reduce  the  fraction,  or  mixed  number,  to  its  simplest 
fractional  form.  Extract  the  cube  root  of  the  numerator  and 
denominator  separately,  if  they  have  exact  roots  ;  if  they  have 
not,  reduce  the  fraction  to  a  decimal,  and  extract  the  root  by 
Case  II. 

EXAMPLES. 

102-104.  What  is  the  cube  root  of  \\\  J  ?  of  TVWjftr  ? 
of  17|? 

105-110.    Find    ^5|;      ^ff;      ^f ;     ^J;      N3/4f ; 


EXAMPLES  INVOLVING  THE  PRINCIPLES  OF  THE  CUBE  ROOT. 

§  143.  It  is  an  established  theorem  of  geometry,  that  all 
similar  solids  are  to  each  other  as  the  cubes  of  their  like  di- 
mensions. 

111.  If  a  cannon-ball,    3    inches  in  diameter,  weigh  8 


276  IN  SOLUTION  AND  EVOLUTION.        [CHAP.  XVL 

pounds,  what  will  a  ball  of  the  same  metal  weigh,  whose 
diameter  is  4  inches? 

By  the  above  theorem,  we  have 

S3  :  43  :  :  8  pounds  :  the  answer. 

112.  The  celebrated  Stockton  gun,  which,  in  bursting, 
proved  so  fatal  to  many  of  our  distinguished  citizens,  is  said 
to  have  carried  a  ball  12  inches  in  diameter,  which  weighed 
238  pounds.     What  ought  to  be  the  diameter  of  another 
ball  of  the  same  metal,  which  should  weigh  32  pounds  ? 

113.  There  are  three  balls,  whose  diameters  are  respect- 
ively 3,  4,  and  5  inches.     What  is  the  diameter  of  a  fourth 
ball,  which  is  equal  in  weight  to  the  three  balls  ? 

114-115.  Which  will  weigh  the  most,  three  balls  whose 
diameters  are  respectively  15,  22,  and  41  inches,  or  three 
balls  of  the  same  metal  whose  diameters  are  respectively  20, 
25,  and  39  inches  ?  What  is  the  diameter  of  a  ball  whose 
weight  is  the  average  of  the  weights  of  the  six  balls  ? 

116.  A  cooper  having  a  cask  40  inches  long  and  32 
inches  at  the  bung  diameter,  wishes  to  make  another  cask 
of  the  same  shape,  which  shall  contain  just  twice  as  much. 
What  will  be  the  dimensions  of  the  new  cask  ? 

117.  What  is  the  side  of  a  cube,  which  will  contain  as 
much  as  a  chest  8  feet  3  inches  long,  3  feet  wide,  and  2  feet 
7  inches  deep  ? 

118.  How  many  cubic  quarter  inches  can  be  made  out 
of  a  cubic  inch  ? 

119.  Required  the  dimensions  of  a  rectangular  box,  which 
shall  contain  20000  solid  inches,  the  length,  breadth,  and 
depth  being  to  each  other  as  4,  3,  and  2. 


§  144.]  ARITHMETICAL  PROGRESSION.  277 

CHAPTER   XVII. 

PROGRESSION. 
ARITHMETICAL   PROGRESSION. 

$  144.  A  SERIES  of  numbers,  which  succeed  each  other 
by  a  common  difference,  is  said  to  be  in  arithmetical  pro- 
gression. When  the  terms  are  constantly  increasing,  the 
series  is  an  arithmetical  progression  ascending  ;  when  con- 
stantly decreasing,  the  series  is  an  arithmetical  progression 
descending.  Thus,  1,  3,  5,  7,  9,  <fec.,  is  an  ascending  arith- 
metical progression ;  and  10,  8,  6,  4,  2,  is  a  descending 
arithmetical  progression. 

The  terms  of  an  arithmetical  progression  may  be  frac- 
tional ;  as, 

i,   1,   H,  2,     2J,  3,     3},  4,     41    &c.; 
J,  f,   1,     1J,   If,  2,     21,  2f,  3,     &c. 

The  first  has  a  common  difference  of  J ;  the  second  has  a 
common  difference  of  ^. 

In  arithmetical  progression,  there  are  five  things  to  be 
considered : 

1.  The  fast  term.  2.  The  last  term.  3.  The  common 
difference.  4.  The  number  of  terms.  5.  The  sum  of  all  the 
terms. 

These  quantities  are  so  related  to  each  other,  that  any 
three  of  them  being  given,  the  remaining  two  can  be 
found.  We  will  demonstrate  two  of  the  most  important 
cases. 

CASE.    I. 

By  our  definition  of  an  ascending  arithmetical  progression, 
it  follows  that  the  second  term  is  equal  to  the  first,  increased 

24 


278  PROGRESSION.  [CHAP.  XVII. 

by  the  common  difference ;  the  third  is  equal  to  the  first, 
increased  by  twice  the  common  difference ;  the  fourth  is 
equal  to  the  first,  increased  by  three  times  the  common  dif- 
ference ;  and  so  on,  for  the  succeeding  terms. 

Hence,  to  find  the  last  term,  when  the  first  term,  the 
common  difference,  and  the  number  of  terms  are  given,  to 
the  first  term  add  the  product  of  the  common  difference  into 
the  number  of  terms,  less  one. 


EXAMPLES. 

1.  What  is  the  10 Oth  term  of  an  arithmetical  progression, 
whose  first  term  is  2,  and  common  difference  3  ? 

2.  What  is  the  50th  term  of  the  arithmetical  progression, 
whose  first  term  is  1,  the  common  difference  J  ? 

3.  A  man  buys  10  sheep,  giving  $1  for  the  first,  $3  for 
the  second,  $5  for  the  third,  and  so  on,  increasing  in  arith- 
metical progression.     What  did  the  last  sheep  cost  at  that 
rate? 

4.  The  first  term  of  an  arithmetical  progression  is  |-,  the 
common  difference  -|-,  and  the  number  of  terms  26.     What 
is  the  last  term  ? 

5.  A  tapering  board,  6  inches  wide  at  the  narrow  end, 
and   12  feet  long,  is  found  to  increase  -J  an  inch  for  every 
foot  in  length.     What  is  the  width  of  the  wide  end  ? 

6.  A  field  of  maize,  consisting  of  50  rows,  has  20  hills  in 
the  first  row,  23  in  the  second  row,  and  so  on,  each  row 
having  three  hills   more  than  the   preceding  row.     How 
many  hills  were  in  the  last  row  ? 

7.  A  person  makes   12  monthly  deposits  in  a  savings 
bank ;  the  first  deposit  consisted  of  $25,  the  second  of  $30, 
the  third  of  $35,  and  so  on  in  arithmetical  progression.     How 
much  did  he  deposit  the  12th  month  ? 


§  144:.]  ARITHMETICAL  PROGRESSION.  279 

CASE    II. 

From  the  nature  of  an  arithmetical  progression,  we  see 
that  the  second  term  added  to  the  next  to  the  last  term  is 
equal  to  the  first  added  to  the  last ;  since  the  second  term 
is  as  much  greater  than  the  first  as  the  next  to  the  last  is 
less  than  the  last.  So  we  infer  that  the  sum  of  any  two 
terms  equidistant  from  the  extremes,  is  equal  to  the  sum  of 
the  extremes. 

Hence,  it  follows  that  the  terms  will  average  just  half  the 
sum  of  the  extremes. 

Therefore,  to  find  the  sum  of  all  the  terms,  when  the  first 
term,  the  last  term,  and  the  number  of  terms  are  giver, 
multiply  half  the  sum  of  the  extremes  by  the  number  of 
terms. 

EXAMPLES. 

8.  The  first  term  of  an  arithmetical  progression  is  2,  the 
last  term  is  50,  and  the  number  of  terms  is  17.    What  is  the 
sum  of  all  the  terms  ? 

9.  The  first  term  of  an  arithmetical  progression  is  13,  the 
last  term  is  1003,  the  number  of  terms  is  100.    What  is  the 
sum  of  the  progression  ? 

10.  A  person  travels  25  days,  going  11  miles  the  first 
day,  135  the  last  day;  the  miles  which  he  travelled  in  the 
successive  days  form  an  arithmetical  progression.     JIow  far 
did  he  go  in  the  25  days  ? 

11.  Bought  7  books,  the  prices  of  which  are  in  arithmeti- 
cal progression.     The  price  of  the  first  was  8  shillings,  and 
the  price  of  the  last  was  28  shillings.     What  did  they  all 
come  to  ? 

12.  What  is  the  sum  of  1000  terms  of  an  arithmetical 
progression,  whose  first  term  is  7  and  last  term  1113  ? 


280  PROGRESSION.  [CHAP.  XVII. 

13.  The  first  term  of  an  arithmetical  progression  is  -|,  and 
the  last  term  365-f,  and  the  number  of  terms  799.  What 
is  the  sum  of  all  the  terms  ? 

14-15.  How  many  strokes  does  the  common  clock  make 
in  12  hours?  How  many  strokes  would  be  made  by  a 
clock  which  continues  to  strike  through  all  the  hours  from 
1  to  24  ? 

16.  A  falling  body  moves  16T^  feet  during  the  first  sec- 
ond, and  144J  feet  during  the  fifth  second.     How  far  did  it 
fall  during  the  5  seconds  ? 

17.  A  person  makes  12  monthly  deposits,  which  are  in 
arithmetical  progression,  in  a  savings  bank.     The  first  de- 
posit is  $20,  the  last  $75.     What  is  the  whole  sum  thus 
deposited  ? 

GEOMETRICAL  PROGRESSION. 

§  145.  A  series  of  numbers  which  succeed  each  other  by 
a  constant  multiplier,  is  called  a  geometrical  progression. 

This  constant  factor,  by  which  the  successive  terms  are 
multiplied,  is  called  the  ratio.  When  the  ratio  is  greater 
than  a  unit,  the  series  is  called  an  ascending  geometrical  pro- 
gression ;  when  less  than  a  unit,  the  series  is  called  a  de- 
scending geometrical  progression.  Thus,  1,  3,  9,  27,  81,  &c., 
is  an  ascending  geometrical  progression,  whose  ratio  is  3. 
And  1,  J,  -j-L,  Jj,  <fec.,  is  a  descending  geometrical  progres- 
sion, whose  ratio  is  J. 

In  geometrical,  as  in  arithmetical  progression,  there  are 
five  things  to  be  considered : 

1.  The  first  term.  2.  The  last  term.  3.  The  common 
ratio.  4.  The  number  of  terms.  5.  The  sum  of  all  the 
ferms. 


§  14:5.]  GEOMETRICAL  PROGRESSION.  281 

These  quantities  are  so  related  to  each  other,  that  any 
three  being  given,  the  remaining  two  can  be  found. 
We  will  demonstrate  two  of  the  most  important  cases. 


CASE    I. 

By  the  definition  of  a  geometrical  progression,  it  follows 
that  the  second  term  is  equal  to  the  first  term  multiplied  by 
the  ratio ;  the  third  term  is  equal  to  the  first  term  multi- 
plied by  the  second  power  of  the  ratio ;  the  fourth  term  is 
equal  to  the  first  term  multiplied  by  the  third  power  of  the 
ratio  ;  and  so  on,  for  the  succeeding  terms. 

Hence,  to  find  the  last  term,  when  the  first  term,  the 
ratio,  and  the  number  of  terms  are  given,  multiply  the  first 
term  by  the  power  of  the  ratio,  whose  exponent  is  one  less  than 
the  number  of  terms. 

EXAMPLES. 

18.  The  first  term  of  a  geometrical  progression  is  1,  the 
ratio  is  2,  and  the  number  of  terms  is  7.     What  is  the  last 
term? 

19.  The  first  term  of  a  geometrical  progression  is  5,  the 
ratio  is  4,  and  the  number  of  terms  9.     What  is  the  last 
term? 

20.  A  person  travelling,  goes  5  miles  the  first  day,  10 
miles  the  second  day,  20  miles  the  third  day,  and  so  on, 
increasing  in  geometrical  progression.      If  he  continue  to 
travel  in  this  way  for  7  days,  how  far  will  he  go  the  last 
day? 

21-23.  A  person  in  business  found  that  he  was  able  to 
double  his  capital  once  in  3  years :  if,  then,  he  commence 
business  with  $1000,  what  will  his  capital  amount  to  at  the 
end  of  the  12th  year  ?  How  much  will  it  amount  to  at  the 

24* 


282  PROGRESSION.  [CHAP.  XVII. 

end  of  the  18th  year  ?     How  much  at  the  end  of  the  24th 
year? 

CASE    II. 

Let  it  be  required  to  find  the  sum  of  all  the  terms  of  the 
geometrical  progression,  2,  6,  18,  54,  162,  486. 

A  second  progression  is,  6,  18,  54,  162,  486,  1458 ;  which 
is  the  result  of  the  multiplication  of  each  term  of  the  first, 
by  the  ratio  3.  The  sum  of  the  terms  of  this  2d  progres- 
sion is  evidently  three  times  as  great  as  the  sum  of  the  terms 
of  the  first  progression ;  while  the  difference  between  the 
sums  of  the  terms  of  these  two  progressions  must  be  (3  —  1) 
=  2  times  the  sum  of  the  terms  of  the  first  progression. 

If  we  omit  the  first  term  of  the  first  progression,  it  will 
agree  with  the  second  progression,  after  omitting  its  last 
term.  Hence,  the  difference  between  the  sums  of  the  terms 
of  these  two  progressions  may  be  found  by  subtracting  2, 
the  first  term  of  the  first  progression,  from  1458,  the  last 
term  of  the  second  progression,  leaving  1456.  And  as  this 
difference  =  2  times  the  sum  of  the  terms  of  the  first  pro- 
gression, dividing  the  difference  by  2  =  3  —  1,  will  give  the 
sum  of  the  terms  required. 

Therefore,  to  find  the  SUM  OF  THE  TERMS  of  a  geometrical 
progression,  when  the  first  term,  the  last  term,  and  the  ratio 
are  given,  divide  the  difference  between  the  first  term  and  the 
last  term  multiplied  by  the  ratio,  by  the  difference  between  the 
ratio  and  1. 

EXAMPLES. 

24.  The  first  term  of  a  geometrical  progression  is  4,  the 
last  term  is  78732,  and  the  ratio  is  3.     What  is  the  sum  of 
all  the  terms  ? 

25.  The  first  term  of  a  geometrical  progression  is  5,  the 


§  145.]  GEOMETRICAL  PROGRESSION.  283 

last  term  is  32*7680,  and  the  ratio  is  4.     What  is  the  sum 
of  all  the  terms  ? 

26.  A  person  sowed  a  peck  of  wheat,  and  used  the  whole 
crop  for  seed  the  following  year  ;  the  produce  of  the  second 
year  again  for  seed  the  third  year,  and  so  on.     If  in  the  last 
year  his  crop  is   1048576  pecks,  how  many  pecks  did  he 
raise  in  all,  allowing  the  increase  to  have  been  in  a  fourfold 
ratio  ? 

NOTE  1.  —  "When  the  ratio  of  a  geometrical  progression  is  le&s  than 
a  unit,  the  first  term  will  be  the  largest,  and  the  last  term  the  least  ; 
the  progression  will,  in  this  case,  be  descending  ;  but  we  may  consider 
the  series  of  terms  in  a  reverse  order  ;  that  is,  we  may  call  the  last 
term  the  first,  and  the  first  the  last,  and  treat  the  progression  as  as- 
cending. 

NOTE  2.  —  If  a  descending  series  be  continued  to  infinity,  the  last 
term  may  be  considered  0. 

What  is  the  sum  of  all  the  terms  of  the  infinite  series  1, 

*,*,*,  Ac.? 

In  this  example,  the  difference  between  the  ratio  and  1,  is  1—  i=£, 
and  the  first  term,  1,  divided  by  -J,  gives  2,  for  the  sum  of  all  the 
tyrms. 

EXAMPLES. 

27.  What  is  the  sum  of  the  infinite  series  1,  i,  -J-,  -Jy,  &c.  ? 

28.  What  is  the  sum  of  the  infinite  series  T^, 
ToVo  «  ,  *c.  ? 

29.  What  is  the  sum  of  the  infinite  series  T2^, 


30-31.  A  ball  falling  from  the  height  of  10  feet,  by  its 
elasticity  bounds  5  feet  ;  and  again  falling,  bounds  2  J  feet, 
and  so  on,"  continuing  to  bound  J  as  high  as  it  fell.  What 
will  be  the  whole  distance  made  in  the  successive  falls  be- 
fore coming  to  a  state  of  rest  ?  And  what  the  whole  dis- 
tance made  by  its  successive  bounds  9 


284 


MENSURATION.  [CHAP.  XVIII. 


8    82     8s 

32.  Find  the  sum  of  the  infinite  series  1,  — ,  — ,  -5-,  &c. 

99     9 

4    4a     48 

33.  Find  the  sum  of  the  infinite  series  1,  -,  — ,  -5-,  &c. 

55      5 


CHAPTER   XVIII. 

MENSURATION. 

§  146.  FOR  the  reason  of  many  of  the  rules  which  we 
shall  give  for  measuring  surfaces  and  solids,  we  shall  refer 
to  the  principles  of  geometry.  The  reference  being  in  all 
cases  to  the  "  Elements  of  Geometry." 


PROBLEM  I. —  To  find  the  area  of  a  rectangle. 


Suppose  ABCD  to  be  a 
rectangle  whose  length  is  5 
feet,  and  width  3  feet. 

If  we  divide  this  rectangle 
into  portions  of  one  square 
foot  each,  by  means  of  lines 
drawn  parallel  to  the  sides  of 


D 


B 


the  rectangle,  we  shall  obtain  15  such  squares  ;  that  is,  the  rectangle 
will  contain  15  square  feet.     In  this  example  there  are  3  strips  of  5 
square  feet  in  each,  or  5  strips  of  3  square  feet  each. 
Hence,  to  find  the  area  of  a  rectangle, 

Multiply  the  length  by  the  width,  and  the  product  will  de- 
note the  number  of  squares  of  the  same  kind  as  the  measure 
used  in  estimating  the  sides  of  the  rectangle.  If  the  measure 
be  feet,  the  product  will  be  square  feet ;  if  inches,  square 
inches,  &c.  (B.  IV.,  Prop.  II.,  Scholium.) 


§  146.] 


MENSURATION. 


NOTE. — When  the  width  of  the  rectangle  is  the  same 
as  its  length,  it  becomes  a  square ;  in  which  case  we  mul- 
tiply the  side  of  the  square  into  itself. 


EXAMPLES. 

1.  How  many  square  feet  in  a  floor  which  is  16  feet  wide 
and  23^-  feet  long  ?     And  how  many  yards  of  carpeting, 
one  yard  wide,  will  cover  the  floor  ? 

2.  In  a  table  5  feet  3  inches  long,  and  3  feet  2  inches 
wide,  how  many  square  inches  ?     And  how  many  square 
feet? 

3.  In  a  rectangular  field  which  is  13  rods  long,  and  7 
rods  wide,  how  many  square  rods  ?     And  what  part  is  it  of 
an  acre  ? 

4.  How  many  square  inches  in  a  square  board  '10J  inches 
on  a  side  ? 

5.  Which  is  the  greater,  a  square  board  of  9  inches  on  a 
side,  or  a  rectangular  one  12  inches  long  and  7^  wide  ? 


PROBLEM  II. — To  find  the  area  of  a  parallelogram. 

Let  ABCD  be  a  parallelo- 
gram having  AB  for  its  base 
and  DE  its  altitude.  If 
from  C  we  draw  CF  perpen- 
dicular to  the  base  A  B, 
meeting  it  produced,  at  the 
point  F,  the  figure  EFCD 

will  be  a  rectangle  equivalent  to  the  parallelogram,  since  the  triangle 
AED  is  obvioush  equal  to  the  triangle  BFC.  The  base  EF  of  the 
rectangle  is  equal  to  AB,  the  base  of  the  parallelogram.  The  area  of 
the  rectangle  is  found  (Paou.  I.)  by  multiplying  its  base  by  its  alti- 
tude ;  and  since  the  parallelogram  is  equal  to  the  rectangle,  and  since 
its  base  and  altitude  are  respectively  equal  to  the  base  and  altitude 


E 


286 


MENSURATION. 


[CHAP.  xvin. 


of  the  rectangle,  it  follows  that  the  area  of  the  parallelogram  may  be 
found  by  multiplying  its  base  by  its  altitude. 
Hence,  to  find  the  area  of  a  parallelogram, 

Multiply  the  base  by  the  altitude. 

PROBLEM  III. — To  find  the  area  of  a  triangle. 

Let  ABC  be  a  triangle, 
having  AB  for  its  base  and 
CD  its  altitude.  By  draw- 
ing CE  parallel  to  the  base 
AB,  and  BE  parallel  to  the 
side  AC,  we  shall  form  a 
parallelogram  ABEC,  evi- 
dently double  the  triangle  ABC.  The  area  of  the  parallelogram  is 
found  (PROB.  II.)  by  multiplying  the  base  AB  into  the  altitude  CD. 
And  as  the  triangle  is  one-half  the  parallelogram,  to  find  its  area, 

Multiply  half  the  base  by  the  altitude. 

NOTE  1.— Either  side  of  the  tri- 
angle  may  be  regarded  as  the 
base,  and  the  altitude  will  be  the 
perpendicular    drawn   from    the 
opposite  angle  to  the  base,  or  to 
the  base  produced.     In  the  an- 
nexed diagram,  the  perpendicu- 
lar meets  the  base  produced.     The  above  rule  applies  equally  well 
in  this  case,  the  area  being  found  by  multiplying  half  the  base  AB 
into  CD. 

When  the  three  sides  of  a  triangle  are  known,  the  area 
may  be  found  as  follows :  From  the  half  sum  of  the  three 
sides,  subtract  separately  each  side,  take  the  square  root  of  the 
continued  product  of  the  three  remainders  and  half  sum,  and 
it  will  give  the  area.  (Geometry,  B.  II.,  Prop.  IX.) 


14 


146.] 


MENSUKATION. 


287 


EXAMPLES. 

6.  What  is  the  area  of  a  triangle  whose  base  is  12  feet, 
and  altitude  3  yards  ? 

7.  What  is  the  area  of  a  triangle  whose  sides  are  respec- 
tively 7,  11,  and  12  feet? 

8.  What  is  the  area  of  a  triangle  whose  base  is  14  rods, 
and  whose  altitude  is  12  rods  ? 

9.  What  is  the  area  of  a  triangle  whose  sides  are  re- 
spectively 13,  14,  and  15  yards? 

10.  In  a  triangular  field,  whose  sides  are  18,  80,  and  82 
feet,  how  many  square  yards  ? 


NOTE  2. — The  area  of  any  figure 
which  is  limited  by  any  number  of 
right  lines,  as  the  field  ABCDEF, 
may  be  found  by  dividing  it  into  tri- 
angles, and  then  computing  each  tri- 
angle separately,  and  taking  their 


PROBLEM  IV. — To  find  the  area  of  a  trapezoid. 

Let  ABCD  be  a  trapezoid, 
having  AB  and  CD  for  the  par- 
allel sides,  CF  for  its  altitude. 
If  we  draw  AC,  it  will  divide 
the  trapezoid  into  two  triangles 
ABC,  CDA.  The  area  of  the 
triangle  ABC  may  be  found  (PROP.  III.)  by  multiplying  half  the  base 
AB  into  the  altitude  CF  ;  and  the  area  of  the  triangle  CDA  is  found 
by  multiplying  half  the  base  CD  into  the  altitude  AE,  or  into  its 
equal  CF.  Hence,  to  find  the  area  of  the  trapezoid,  which  is  the 
sum  of  the  two  triangles, 

Multiply  half  iln>  .??/???,  of  the  two  parallel  sides  by  the 
attitude. 


288  MENSURATION.  [CHAP.  XVIII. 

NOTE. — This  rule  has  a  fine 
application  in  measuring  a 
tapering  board,  as  ABCD. 
In  this  case  half  the  sum  of 
the  parallel  sides,  AD  and 
BC,  is  found  by  measuring 

the  width  GH  at  the  middle  of  the  board.     This  average  width  GH 
being  multiplied  by  the  length  EF,  will  give  its  area. 

EXAMPLES. 

11.  If  the  parallel  sides  of  a  trapezoidal  garden  are  re- 
spectively 4  and  6   rods;  and  the  perpendicular  distance 
between  these  sides  is  8  rods,  how  many  square  rods  in  the 
garden  ? 

12.  How  many  square  feet  in  a  tapering  board  16  feet 
long,  measuring  15  inches  wide  at  one  end,  and  10  inches  at 
the  other  ? 

PROBLEM  Y. — The  diameter  of  a  circle  being  given,  to  find 
its  circumference. 

If  the  diameter  of  a  circle  is  taken  as  a  unit,  the  circumference  will 
be  3*1415926,  nearly.  The  exact  value  of  the  ratio  of  the  circumfer- 
ence to  the  diameter  has  never  been  found.  Its  approximate  value 
has  been  extended  to  more  than  200  places  of  decimals.  (Geometry, 
B.  V.,  Prop.  XIII,  Scholium.) 

Hence,  when  the  diameter  of  a  circle  is  known,  to  find  its  circum- 
ference, 

Multiply  the  diameter  by  3*1416. 
EXAMPLES. 

13.  What  is  the  circumference  of  the  earth,  on  the  sup- 
position that  it  is  8000  miles  in  diameter  ? 

14.  Suppose  a  cart  wheel  be  4ft.  9m.  in  diameter,  over 
what  distance  would  it  pass  in  making  8  revolutions  ? 


§  146.] 


MENSURATION. 


289 


15.  The  hoop  you  drive  is  3ft.  Win.  in  diameter.  How 
many  times  will  it  revolve  in  being  trundled  to  school,  half 
'a  mile  distant  ? 

PROBLEM  VI. — To  find  the  area  of  a  circle,  when  its  diam- 
eter is  known. 

RULE. 

Multiply  the  circumference  by  one-fourth  of  the  diameter. 
Or,  what  is  equivalent,  multiply  the  square  of  the  diameter 
by  0-7854  =  J  o/3'1416.  (Geometry,  B.  V.,  Prop.  XI.) 


NOTE. — If  a  circle  be  inscribed  in  a  square, 
its  area  will  be  to  the  area  of  the  square  as 
0-7854  is  to  1. 


EXAMPLES. 

16.  How  many  acres  in  a  circle  one  mile  in  diameter  ? 
NOTE. — In  a  square  mile  there  are  640  acres. 

17.  Which  is  the  greater  area,  a  circle  5  feet  in  diameter, 
or  the  sum  of  the  areas  of  two  other  circles,  the  one  being 
4  feet  in  diameter  and  the  other  3  feet  ? 

From  the  preceding  rule  we  may 
deduce  a  simple  method  of  rinding 
the  area  comprised  between  the 
circumferences,  of  two  concentric 
circles,  which  area  is  the  difference 
between  two  circles. 

The  area  of  the  circle  whose  di- 
ameter is  AB,  is  found  by  multiply- 
ing its  square  by  0'7854.  And  the 
circle  whose  diameter  is  DE,  is 
found  by  multiplying  the  square  of 

25 


MENSUJRATIOX.  [CHAP  XVIII. 

this  diameter  by  0'7854.  Hence,  the  difference  of  these  areas  is 
equal  to  the  difference  of  the  squares  of  the  diameters  multiplied  by 
0-7854. 

PROBLEM  VII. — To  find  the  volume  of  a  prism,  or  of  a. 
cylinder. 

RULE. 

Multiply  the  area  of  tho  base  by  the  altitude.     (Geometry, 
B.  VII.,  Prop.  XI.) 

EXAMPLES.        , 

1 8.  How  many  cubic  feet  in  a  rectangular  stick  of  timber 
10  inches  by  12  inches,  and  36  feet  long  ? 

19.  In  a  cylindrical  log  14  feet  long,  and  14  inches  in 
diameter,  how  many  cubic  feet  ? 

20.  How  many  cubic  inches  hi  a  round  bar  of  iron  20 
feet  long,  and  J  of  an  inch  in  diameter  ? 

PROBLEM  VIII. — To  find  the  volume  of  a  pyramid,  or  of 
a  cone. 

RULE. 

Multiply  the  area  of  the  base  by  one-third  the  altitude. 
(Geometry,  B.  VII.,  Prop.  XVII. ;  and  B.  VIII.,  Prop.  V.) 

EXAMPLES. 

21.  The  Egyptian  pyramid  Cheops  covers  a  square  of 
763|-  feet  on  a  side,  and  is  480  feet  perpendicular  height. 
How  many  cubic  feet  does  it  contain  ? 

22.  Suppose  the  mast  of  a  ship  to  be  a  regular  cone  87 
feet  long,  and  2  feet  in  diameter  at  its  base,  how  many  <*ubic 
feet  will  it  contain  ? 


§  146.]  MENSURATION.  291 

PROBLEM  IX. — To  find  the  surface  of  a  sphere,  when  its 
diameter  is  given. 


RULE. 


Multiply  the  square  of  the  diameter  by  3  '141 6.     (Geome- 
try, B.  VIII.,  Prop.  XIII.,  Schol.) 

EXAMPLES, 

23.  How  many  square  miles  on  the  surface  of  the  earth, 
on  the  supposition  that  it  is  an  exact  sphere  of  8000  miles 
in  diameter  ? 

NOTE. — In  order  to  obtain  a  value  true  to  a  unit,  we  must  use,  for 
our  multiplier,  3-14159265,  instead  of  3-1416. 

24.  How  many  superficial  inches  has  a  ball  6  inches  in 
diameter? 

PROBLEM    X. — To  find  the  volume    of  a  sphere,   when  its 
diameter  is  given. 

RULE. 

Multiply  the  cube  of  the  diameter  by  0'5236,  which  is  ^  of 
3-1416.     (Geometry,  B.  VIII.,  Prop.  XIII.,  Schol.) 

EXAMPLES. 

25.  How  many  cubic  inches  in  a  ball  6  inches  in  diameter  ? 

NOTE. — Compare  the  number  of  superficial  inches  and  of  cubic 
inches  in  a  sphere  6  inches  in  diameter. 

26.  How  many  cubic  inches  in  a  ball  of  the  celebrated 
Stockton  gun,  the  diameter  of  which  is  12  inches? 


292  MENSURATION.  [CHAP.  XVIIL 

The  following  table  of  multipliers  will  be  found  very  convenient 
for  solving  nearly  all  problems  which  can  arise  in  mensuration  of  cir- 
cles and  spheres. 


TABLE    OF    MULTIPLIERS. 

1.  Radius  of  a  circle  X  6-28318531  =  Circumference. 

2.  Square  of  the  radius  of  a  circle  X  3-14159265  =  Area. 

3.  Diameter  of  a  circle  X  3-14159265  =  Circumference. 

4.  Square  of  the  diameter  of  a  circle  X  0-78539816  =  Area. 

5.  Circumference  of  a  circle  X  0-15915494  =  Radius. 

6.  Circumference  of  a  circle  X  0-31830989  =  Diameter. 

7.  Square  root  of  area  of  a  circle  X  0-56418958  =  Radius. 

8.  Square  root  of  area  of  a  circle  X  1*12837917  =  Diameter 

9.  Radius  of  circle  X  1'73205081  =  Side  of  inscribed  equilateral  triangle. 

10.  Side  of  inscribed  equilateral  triangle  X  0-57735027  =  Radius  of  circle. 

11.  Radius  of  a  circle  X  1-41421356  =  Side  of  inscribed  square. 

12.  Side  of  inscribed  square  X  0  70710678  =  Radius. 

13.  Square  of  radius  of  a  sphere  X  12-56637061  =  Surface. 

14.  Cube  of  radius  of  a  sphere  X  4-18879020  =  Volume. 

15.  Square  of  diameter  of  a  sphere  X  3-14159265=  Surface. 

16.  Cube  of  diameter  of  a  sphere  X  0-52359878  =  Volume. 

17.  Square  of  circumference  of  a  sphere  X  0-31830989  =  Surface. 

18.  Cube  of  circumference  of  a  sphere  X  0-01688686=  Volume. 

19.  Square  root  of  surface  of  a  sphere  X  0-28209479  =  Radius. 

20.  Square  root  of  surface  of  a  sphere  X  0-56418958=  Diameter. 

21.  Square  root  of  surface  of  a  sphere  X  1 '77245385  =  Circumference. 

22.  Cube  root  of  volume  of  a  sphere  X  0-62035049  =  Radius. 

23.  Cube  root  of  volume  of  a  sphere  X  1-24070098  =  Diameter. 

24.  Cube  root  of  volume  of  a  sphere  X  3-89777707  =  Circumference. 

25.  Radius  of  a  sphere  X  1-15470054  =  Side  of  inscribed  cube. 

26.  Side  of  inscribed  cube  X  0-86602540  =  Radius. 


PROBLEM  XI. — To  find  the  volume  of  a  frustum  of  a  pyr- 
amid, or  of  a  cone. 

RULE. 

.Find  a  mean  proportional  between  the  area  of  the  two 
bases,  to  which  add  the  sum  of  the  bases,  and  multiply  the 
result  by  one-third  the  altitude  of  the  frustum. 

Suppose  a  cistern  in  the  form,  of  a  frustum  of  a  cone  to 


§  146.]  MENSURATION.  293 

be  9  feet  deep,  having  for  diameters  10  feet  and  6  feet.    How 
many  cubic  feet  will  it  contain  ? 

102XO-7854=100XO-7854=area  of  one  base. 
62XO-7854=  36X0-7854=   "         other" 

60XO'7854=mean  proportion  between  bases. 

196XO-7854=sum. 
And  196XO-7854X £  of  9=461-8152  cubic  feet  for  its  volume. 

EXAMPLES. 

2*7.  Suppose  a  measure  to  be  in  the  form  of  a  frustum  of 
a  regular  cone.  If  its  top  diameter  is  6  inches,  and  the  bot- 
tom diameter  9  inches,  and  it  is  12  inches  deep,  how  many 
cubic  inches  will  it  contain  ?  and  how  many  beer  gallons  of 
282  cubic  inches  each  ? 

28.  There  is  a  stick  of  timber  in  the  form  of  the  frustum 
of  a  regular  pyramid,  which  is  30  feet  long,  and  30  inches 
square  at  one  end  and  13  inches  square  at  the  other.  How 
many  cubic  feet  does  it  contain  ? 

PROBLEM  XII. —  To  find  the  area  of  an  ellipse. 

NOTE. — A  line  drawn  through 

the  centre  of  an  ellipse  is  called  ^___— — 

its  diameter.  The  longest  diam- 
eter is  called  the  transverse  diam- 
eter ;  the  shortest  is  called  the 
conjugate  diameter.  Thus  AB  is 
the  transverse  diameter,  and  CD 
is  the  conjugate  diameter. 

The  area  of  an  ellipse  may  be  found  by  this 

RULE. 

Multiply  the  product  of  the  transverse  and  conjugate  di- 
ameters by  0'7854. 

25* 


PROMISCUOUS   QUESTIONS.          [CHAP.  XVIII. 


EXAMPLES. 

29.  How  many  square  feet  in  the  surface  of  an  elliptical 
pond,  whose  transverse  diameter  is  100  feet,  and  conjugate 
diameter  60  feet  ? 

30.  How  many  square  inches  in  an  elliptical  table  whose 
transverse  diameter  is  5  feet  3  inches,  and  conjugate  diam- 
eter 3  feet  6  inches  ?     And  how  many  square  feet  ? 


NOTE  1.  —  If  an  ellipse  be  inscribed 
in  a  rectangle,  its  area  will  be  to 
the  area  of  the  rectangle  as  0*7854 


is  to  1. 


NOTE  2. — We  also  infer  that,  if  a 
circle  be  inscribed  in  an.  ellipse,  and 
another  circle  be  circumscribed  about 
the  same  ellipse,  the  ellipse  is  a  mean 
proportional  between  the  areas  of 
the  two  circles;  that  is,  we  shall 
have,  area  of  inscribed  circle  is  to 
the  area  of  ellipse,  as  area  of  el- 
lipse is  to  the  area  of  circumscribed 
circle. 


PROMISCUOUS   QUESTIONS. 

§147.  31.  Suppose  I  purchase  $1200  worth  of  goods, 
J  of  which  is  on  a  credit  of  3  months,  J  on  a  credit  of  6 
months,  and  the  remaining  ^  on  a  credit  of  9  months. 
How  much  ready  money  ought  to  pay  the  purchase,  inter- 
est being  7  per  cent.  ? 

32.  In  the  above   example,  by  the  principles  of  equation 


§  14:7.]  PROMISCUOUS   QUESTIONS.  295 

of  payments,  how  much  credit  ought  I  to  have  on  the  whole 
sum  of  $1200? 

33.  Now,  what  is  the  present  worth  of  $1200  due  at  the 
end  of  6  months,  interest  being  7  per  cent.  ? 

34.  I  employed  A.  and  B.  to  ditch  my  meadow.     A.  was 
to  receive  87 -J  cents  per  rod,  and  B.  was  to  have  112  J  cents 
per  rod  ;  each  wrought  until  his  wages  amounted  to  $50. 
What  was  the  amount  of  ditch  dug  by  both  ? 

35.  Three  merchants,  A.,  B.,  and  C.,  enter  into  partner- 
ship.    A.    advances   $1200,  B.  $800,  and  C.   $600.     A. 
leaves  his  money  8  months,  B.  10  months,  and  C.  14  months 
in  the  business.     They  gain  $500.     What  is  the  share  of 
each  ? 

36.  A.  and  B.  have  the  same  income  :  A.  saves  £  of  his, 
but  3.,  by  spending  $120  per  annum  more  than  A.,  at  the 
end  of  10  years  finds  himself  $200  in  debt.     What  was  the 
income  ? 

37.  Suppose  a  book  to  contain  365  pages,  averaging  40 
lines  of  10  words  each  on  each  page.     How  many  words 
would  the  book  contain  ? 

38.  There  are  31173   verses  in  the  Bible.     How  many 
days  will  it  require  to  read  it  through,  if  30  verses  are  read 
daily  ? 

39.  After  expending  i  of  my  money,  and  -J-  of  the  re- 
mainder, I  had  remaining  $72.     How  much  had  I  at  first  ? 

40.  If  I  sell  cloth  at  $1'50  per  yard,  and  gain  25  per 
cent.,  how  ought  I  to  have  sold  it  so  as  to  lose  20  percent.  ? 

41.  Sold  cloth  at  $T50  per  yard,   and  gained  25  per 
cent.     What  should  I  have  lost  per  cent.,  if  I  had  sold  it  at 
$0'96  per  yard  ? 

42.  If  I  buy  cloth  at  $1'20  per  yard,  how  must  I  sell  it 
so  as  to  gain  25  per  cent.  ? 

43.  A  merchant  has  to  make    the  following   payments 


296  PKOMISCUOUS  QUESTIONS.        [CHAP.  XVIH. 

at  three  different  periods:  $2832  in  3  months,  $2560  in  9 
months,  and  $1450  in  16  months.  The  creditor  wishes  to 
receive  the  whole  sum  of  $6842  at  once.  When  ought  the 
payment  to  be  made  ? 

44.  A  father  gives  to  his  five  sons  $1000,  which  they 
are  to  divide  according  to  their  ages,  so  that  each  elder  son 
shall  receive  $20  more  than  his  next  younger  brother.  What 
is  the  share  of  the  youngest  ? 

45.  A  company  of  90   persons  consists  of  men,  women, 
and  children.     The  men  are  4  in  number  more  than  the  wo- 
men, the  children  10  more  than  the  adults.    How  many  men, 
women,  and  children  are  there  in  the  company  ? 

46.  The  common-school  fund  for  the  State  of  New  York 
was  $1975093-15  in  1843,  and  during  the  same  year  there 
were  in  the  State  677995  children  between  the  ages  of  5 
and  16  years.    How  much  would  the  above  fund  amount  to 
per  child  ? 

47.  Two  persons,  A.  and  B.,  being  on  opposite  sides  of 
a  fish-pond,  which  is  536  feet  in  circumference,  begin  to 
walk  around  it  at  the  same  time,  both  in  the  same  way :  A. 
goes  at  the  rate  of  31  yards  per  minute,  and  B.  at  the  rate 
of  34  yards  per  minute.     In  what  time  will  B.  overtake  A.  ? 
And  how  far  will  A.  have  walked  ? 

48.  How  much  money  which  is   23  per  cent,  below  par 
will  pay  a  debt  of  $450  ? 

49.  A.,  B.,  and  C.  commence  trade  with  $3053'25,  and 
gain  $610*65.     A.'s  stock,  together  with  B.'s,  is  to  the  sum 
of  A.'s  and  C.'s  stock  as  5  to  7  ;  and  C.'s  stock,  diminished 
by  B.'s,  is  to  C.'s  increased  by  B.'s  as  1  to  7.     What  was 
each  one's  part  of  the  gain  ? 

50.  A.,  on  preparing  for  a  voyage  to  California,  purchased 
of  B.  specie-dollars,  at  a  premium  of  3  per  cent.,  to  be  paid 
in  1 8  months,  with  interest  at  5  per  cent,  per  annum,  which 


§  147.]  PROMISCUOUS  QUESTIONS.  297 

was  to  be  added  into  the  note.     The  amount  of  the  note  was 
$22145.     How  many  specie-dollars  did  he  receive  ? 

51.  Sold  goods  to  the  amount  of  $3000,  one  half  to  be 
paid  in  3  months,  the  other  half  in  6  months.     How  much 
ought  to  be   discounted  for  ready  money,  when  money  is 
worth  7  per  cent,  per  annum  ? 

52.  The  Falls  of  Niagara  have  receded  nearly  50  yards 
within  the  last  40  years.     How  long,  at  this  rate,  has  it  ta- 
ken them  to  recede  from  Queenstown,  7  miles  below  their 
present  site  ? 

53.  It  is  found  that  the  diameter  of  every  circle  is  to 
its  circumference  very  nearly  in  the  ratio  of  113  to  355. 
What,  then,  is  the  earth's  circumference,  its  diameter  being 
7912  miles? 

54.  How  many  men  must  be  employed  to  perform  in 
26  days  what  60  men  could  do  in  39  days  ? 

55.  If  72  sheep  can  graze  in  a  field  36  days,  how  long 
might  144  sheep  graze  equally  well? 

56.  If  a  locomotive  pass  from  Albany  to  Schenectady, 
a  distance  of  17  miles,  in  45  minutes,  what  time  will  it  re- 
quire, at  the  same  rate,  to  go  from  Schenectady  to  Utica,  a 
distance  of  78  miles  ? 

57.  If  A.  and  B.,  with  C.  working  half  time,  can  build 
a  wall  in  21  days  ;  B.  and  C.,  with  D.  working  half  time,  in 
24  days  ;  C.  and  D.,  with  A.  working  half  time,  in  28  days  ; 
D.  and  A.,  with  B.  working  half  time,  in  32  days  ;  in  what 
time  would  it  be  built  by  all  together,  and  by  each  alone  ? 

58.  One- third  of  a  quantity  of  flour  being  sold  to  gain 
a  certain  rate  per  cent.,  one-fourth  to  gain  twice  as  much 
per  cent.,  and  the  remainder  to  gain  three  times  as  much 
per  cent. ;  it  is  required  to  determine  the  gain  per  cent,  on 
each  part,  the  gain  upon  the  whole  being  20  per  cent. 

59.  A   servant  draws  off   one  gallon  each   day,   for  5 


298  PROMISCUOUS  QUESTIONS.          [CHAP  XVIII. 

days,  from  a  cask  containing  10  gallons  of  wine,  each  time 
supplying  the  deficiency  by  the  addition  of  a  gallon  of  wa- 
ter ;  and  then,  to  escape  detection,  he  again  draws  off  5 
gallons,  each  time  supplying  the  deficiency  by  a  gallon  of 
wine.  It  is  required  to  find  how  much  water  still  remains 
in  the  cask. 

60.  Find  the    four  smallest  numbers,  such    that  when 
each  is  divided  successively  by  2,  3,  4,  5,  6,  7,  8,  and  9,  the 
remainders  shall  in  each  case  be  1. 

61.  Find  the    four  smallest  numbers,  such  that  when 
each  is  divided  by  2  the  remainders  shall  be  1 ;  when  di- 
vided by  3,  the  remainders  shall  be  2 ;  when  divided  by  4, 
the  remainders  shall  be  3  ;  and  so  on,  until  divided  by  9, 
when  the  remainders  shall  be  8.     In  each  case  the  remain- 
der being  1  less  than  the  divisor. 

62.  If  750  men  require  22500  rations  of   bread  for  a 
month,  how  many  rations  will  a  garrison  of  1200  men  re- 
quire for  the  same  time  ? 

63.  How  many  yards   of  paper  that  is  ,30  inches  wide 
will  hang  a  room  that  is  20  yards  in  circuit,  and  9  feet  high  ? 

64.  There  is  a  ladder  with  a  hundred  steps  :  on  the  first 
step  is  seated  1  pigeon ;  on  the  second  2  ;  on  the  third  3  ; 
and  so  on,  increasing  by  one  for  each  step.     How  many 
pigeons  were  seated  on  the  ladder  ? 

65.  If  9  porters  drink  in  8  days  12  casks  of  wine,  how 
many  casks  will  serve  24  porters  for  30  days  ? 

66.  If  3  pounds  of  tea  be  worth  4  pounds  of  coffee,  and 
6  pounds  of  coffee  be  worth  20  pounds  of  sugar,  how  many 
pounds  of  sugar  may  be  had  for  9  pounds  of  tea  ? 

67.  If  48  feet  of  Cremona  equal  56  English  feet,  3 9 '3 71 
English  inches  equal  one  metre  of  France,  how  many  Cre- 
mona feet  is  the  French  metre  ? 

68.  If  a  certain  number  of  men  can  throw  up  an  intrench- 


§  14:7.]  PROMISCUOUS  QUESTIONS.  299 

ment  in  10  days,  when  they  work  6  hours  per  day,  in  what 
time  would  they  do  it  if  they  work  8  hours  per  day  ? 

69.  If  12  men  reap  a  field  of  wheat  in  3  days,  in  what 
time  can  the  same  work  be  done  by  25  men  ? 

70.  A  ship's  crew  of  300  men  were  so  supplied  with 
provisions  for  12  months,  that  each  man  was  allowed  30 
ounces  per  day ;  but  after  sailing  6  months,  they  find  that 
it  will  take  9  months  more  to  finish  their  voyage,  and  50  of 
the  crew  have  been  lost.     Required  the  daily  allowance  of 
each  man  for  the  last  9  months. 

71.  A.,  B.,  and  C.  are  to  share  $1000  in  the  ratio  of  the 
numbers  3,  4,  and  5  ;  but  C.  dying,  it  is  required  to  divide 
the  whole  sum  equitably  between  A.  and  B. 

72.  The    expense  of    repairing  a  school-house  to   the 
amount  of  $600  is  paid  by  three  individuals,  A.,  B.,  and  C., 
in  the  ratio  of  their  nearness  to  it.     What  did  each  pay,  if 
we  suppose  A.  lived  1  mile  distant,  B.  2  miles,  and  C.  3 
miles  ? 

73.  A  merchant  bought  a  piece  of  cloth  for  240  dollars, 
and  sold  a  portion,  exceeding  three-fourths  of  the  whole  by 
2  yards,  at  a  profit  of  25  per  cent.     He  afterwards  sold  the 
remainder  at  such  a  price  as  to  clear  60  per  cent,  by  the 
whole  transaction ;  and  had  he  sold  the  whole  quantity  at 
the  latter  price  he  would  have  gained  175  per  cent.     How 
many  yards  were  there  in  the  piece  ? 

74.  The  whole  number  of  volumes  in  the  common-school 
libraries  of  New  York,  in  1843,  was  874865.     What  would 
be  their  value  at  37 J  cents  per  volume  ? 

75.  The  whole  number  of  children  taught  in  New  York 
during  the  year  1843,  was  657782,  and  the  whole  number 
of  schools  was  10860.     How  many  scholars  on  an  average, 
would  each  school  consist  of  ? 

76.  Suppose  the  Erie  Canal  to  be  60  feet  wide,  and  6 


300  PROMISCUOUS  QUESTIONS.  [CHAP.  XVHI. 

feet  deep,  how  many  miles  in  length  will  it  require  to  make 
one  cubic  mile  of  water  ? 

77.  A  person  owning  £  of  a  copper-mine,  sells  f  of  his 
interest  in  it  for  $1800.     What,  at  this  rate,  is  the  value  of 
the  whole  ? 

78.  Suppose  I  buy  a  certain  lot  of  oranges  at  3  cents 
a  piece,  and  as  many  more  at  5  cents  a  piece,  and  sell  them 
at  4  cents  a  piece  ;  do  I  gain  or  lose  by  the  operation  ? 

79.  Suppose  I  buy  a  certain    number  of   oranges  at  3 
for  one  cent,  and  as  many  more  at  5  for  one  cent,  and  sell 
them  at  4  for  one  cent ;  do  I  gain  or  lose  by  the  operation  ? 

80.  Suppose  I  expend  a  certain  sum  of  money  for  oranges 
at  J  of  a  cent  a  piece,  and  another  equal  sum  for  another 
lot  of  oranges  at  -J  of  a  cent  a  piece,  and  sell  them  at  ^  of  a 
cent  a  piece,  do  I  gain  or  lose  by  the  operation  ? 

81.  Suppose  I  expend  a  certain  sum  of  money  for  oranges 
at  3  cents  a  piece,  and  another  equal  sum  for  another  lot 
at  5  cents  a  piece  ;  how  much  do  I  gain  on  each  cent  ex- 
pended, if  I  sell  them  at  4  cents  a  piece  ? 

82.  If  A.  can  do  a  piece  of  work  in  3  days,  B.  in  4  days, 
and  C.  in  5  days,  how  many  times  longer  will  it  take  B.  to 
do  it  alone,  than  it  will  take  A.  and  C.  together  to  do  it  ? 

83.  If  A.  can  accomplish  a  piece  of  work  in  J  of  a  day, 

B.  in  1  of  a  day,  and  C.  in  J  of  a  day,  how  many  times 
longer  will  it  take  B.  to  do  it  alone,  than  it  will  take  A.  and 

C.  together  to  do  it  ? 

84.  What  is  the  shortest  piece  of  cloth  which  shall  be 
at  the  same  time  an  even  number  of  yards,  an  even  number 
of  Ells  Flemish,  an  even  number  of  Ells  English,  and  an 
even  number  of  Ells  French  ? 

85.  A  man  died,  leaving  $1000,  to  be  divided  between 
his  two  sons,  one   14  and  the  other  18  years  of  age,  in 
such  a  proportion,  that  the  share  of  each  being  put  to  in- 


$  147.]  PROMISCUOUS  QUESTIONS.  301 

terest  at  6  per  cent.,  should  amount  to  the  same  sum  when 
they  should  arrive  at  the  age  of  21.  What  did  each  one 
receive  ? 

86.  Divide  $100  between  A.,  B.,  and  C.,  so  that  B.  may 
have  $3  more  than  A.,  and  C.  $4  more  than  B.     How  much 
must  each  one  have  ? 

87.  A.  can  do  a  piece  of  work  in  4  days,  and  B.  can  do 
the  same  in  3  days.     How  long  would  it  take  both  together 
to  do  it  ? 

88.  A  person  wishes  to   dispose  of  his  horse  by  lottery. 
If  he  sells  the  tickets  at  $2  each,  he  will  lose  $30  on  his 

'horse ;  but  if  he  sells  them  at  $3  each,  he  will  receive  $30 
more  than  his  horse  cost  him.  What  is  the  value  of  the 
horse,  and  the  number  of  tickets  ? 

89.  Thomas  sold  150  pine -apples  at  33  J  cents  a  piece, 
and  received  the  same  amount  of  money  that  Henry  did 
for  watermelons  at  25  cents  a  piece.     How  much  money 
did  each  receive,  and  how  many  melons  did  Henry  sell  ? 

90.  A  man  bought  apples  at  5  cents  a  dozen,  half  of 
which  he  exchanged  for  pears,  at  the  rate  of  8  apples  for 
5  pears ;  he  then  sold  all  his  apples  and  pears  at  a  cent 
a  piece,  and  thus  gained  19  cents.     How  many  apples  did 
he  buy,  and  how  much  did  they  cost  ? 

91.  A  person  expended  $23*40  for  eggs.      With  one- 
half  of  his  money  he  purchased  a  lot  at  13  cents  per  dozen  ; 
with  the  other  half  of   his  money   he  purchased  another 
lot  at  9  cents  per  dozen.     He  afterwards  sold  them  all  to- 
gether at  11  cents  per  dozen.     Did  he  gain  or  lose  by  the 
operation  ? 

92.  Divide  $1200  between  A.  and  B.,  so  that  A.'s  share 
may  be  to  B.'s  as  2  to  7. 

93.  A  gentleman  spends  ^  of   his  yearly  income    for 
board  and  lodging,  f  of  the  remainder  for  clothes,  and  f  of 

26 


302  PROMISCUOUS  QUESTIONS.          [CHAP.  XVIIL 

the  residue  he  bestows  for  charitable  purposes,  and  saves 
8100  yearly.     What  is  his  income  ? 

94.  If  I  buy  an  article  for  $4,  and  sell  it  for  $5,  how 
much  per  cent,  do  I  gain  ? 

95.  If  I  give  $5  for  an  article,  and  sell  it  for  $4,,  how 
much  per  cent,  do  I  lose  ? 

96.  What  is  the  interest  of  $175  for  3  months,  at  6  per 
cent.  ? 

97.  How  many  yards  of  Brussels  carpeting,  which   is  J 
of  a  yard  wide,  will  it  require  to  cover  a  floor  18  feet  by  20 
feet? 

98.  Admitting  the  velocity  of  a  cannon-ball  to  be  1600 
feet  per  second,  what  time,  at  this  velocity,  would  it  require 
to  move  95  millions  of  miles,  which  is  the  distance  from  the 
earth,  to  the  sun,  counting  365J  days  to  the  year. 

99.  The  Winchester  bushel  measure  is  of  a  cylindric  form, 
8  inches  deep,  and  18^  inches  in  diameter,  containing  2150f 
cubic  inches.     What  must  be  the  size  of  a  cubical  box  which 
shall  contain  the  same  quantity  ? 

100.  The  clocks  of  Italy  go  on  to  24  hours ;  then  how 
many  strokes  do  they  strike  in  one  revolution  of  the  index  ? 

101.  There  is  an  island  20  miles  in  circumference,  and 
three  men,  A.,  B.,  and  C.,  start  from  the  same  point,  and 
travel  the  same  way  around  it ;  A.  goes  3  miles  per  hour, 
B.  goes  7  miles  per  hour,  and  C.  goes  11  miles  per  hour. 
In  what  time  will  they  all  be  together  ? 

102.  What  is  the  discount  of  $175  for  3  months,  at  6  per 
cent.  ? 

103.  If  a  ship  and  its  cargo  are  worth  $30000,  and  the 
cargo  is  worth  5  times  as  much  as  the  ship,  what  is  the 
value  of  the  cargo  ? 

104.  What  is   the   difference  between  six  and  one  half 
times  7,  and  seven  and  one  half  times  6  ? 


§   147.]  PROMISCUOUS  QUESTIONS.  303 

105.  Three  persons,  A.,  B.,  and  C.,  form  a  partnership  : 

A.  furnishes  $1000,  B.  $600,  and  C.  $450;  at  the  end  of 

6  months,  C.  withdraws  his  capital,  but  no  dividend  is  made 
until  the  end  of  the  year,  when  it  is  found  that  the  firm 
has  gained  $244*16.     How  is  this  gain  to  be  divided  be- 
tween the  partners  ? 

106.  Three  persons,  A.,  B.,  and  C.,  engage  to  build  a 
certain  piece  of  wall  for  $244*16.     While  A.  can  build  10 
rods,  B.  can  build  but  6,  and  C.  but  4^.     When  the  wall 
is  half  completed,  C.  ceases  to  labor  upon  it,  and  A.  and 

B.  finish  it.     What  part  of  the  $244-16  ought  each  to  re- 
ceive ? 

107.  A.  and  B.  together  can  build  a  wall  in  4  days,  A. 
and  C.  can  together  build  it  in  5  days,  B.  and  C.  can  to- 
gether build  it  in  6  days.     What  time  would  it  require  for 
all  together  to  accomplish  it  ? 

108.  A  note  of  $10000  given  January  1st,  1840,  has  re- 
ceived the  following  indorsements  :  January  1st,  1841,  in- 
dorsed $2952-28;  January  1st,   1842,  indorsed  $2952'28; 
January   1st,   1843,   indorsed  $2952 -28.      How  much  re- 
mained due  January  1st,  1844,  interest  being  computed  at 

7  per  cent.  ? 

109.  Two  hunters,  A.  and  B.,  kill  a  deer,  whose  weight 
they  are  desirous  of  knowing.     For  this  purpose  they  rest 
a  stick  across  the  limb  of  a  tree  ;  then  suspending  the  deer 
at  the  shorter  extremity,  they  find  that  its  weight  is  just 
counterpoised  by  the"  weight  of  A.,  who  suspends  himself 
by  his  hands  at  the  other  extremity.     Without  changing 
the  point  of  support  of  the  stick,  they  take  the  deer  from 
the  shorter  extremity,  and  suspend  it  at  the  longer  extrem- 
ity of  the  stick,  when  it  was  found  to  be  exactly  balanced 
by  B.'s  weight,  when  suspended  to  the  shorter  extremity 
of  the  stick.     Now,  supposing  A.  to  weigh  147  pounds,  and 


304:  PROMISCUOUS  QUESTIONS.         [CHAP.  XVIII. 

B.  to  weigh  192  pounds,  what  must  have  been  the  weight 
of  the  deer  ? 

NOTE. — By  the  principle  of  the  lever,  when  different  weights  at 
its  extremities  balance  each  other,  they  are  to  each  other  inversely 
as  the  lengths  of  the  arras  to  which  they  are  attached.  Hence,  in 
the  first  experiment,  we  know  that  the  weight  of  A.  is  to  the  deer's 
weight,  as  the  shorter  arm  is  to  the  longer  arm.  In  the  second  ex- 
periment, the  deer's  weight  is  to  B.'s  weight,  as  the  shorter  arm  is 
to  the  longer  arm.  Consequently,  A.'s  weight  is  to  the  deer's  weight» 
as  the  deer's  weight  is  to  B.'s  weight. 


APPENDIX. 


CHAPTER  I. 

§  1.  WHAT  is  a  unit  ?  What  is  a  number  ?  Give  an  instance 
of  a  number.  What  is  an  abstract  number  ?  If  a  number  be  not 
abstract,  what  is  it  ?  What  is  the  difference  between  an  abstract 
and  a  denominate  number  ?  Give  examples  of  each  kind.  What 
does  denominate  mean  ?  If  I  say  there  are  365  days  in  a  year, 
what  kind  of  a  number  do  I  make  use  of?  Why?  How  will 
you  use  the  number  365  to  make  it  an  abstract  number  ? 

$  2.  Of  what  does  Arithmetic  treat  ?  What  is  it  as  a  science  ? 
As  an  art  ?  How  many  methods  of  expressing  numbers  are  there  ? 
What  are  the  different  methods  ? 

CHAPTER  II. 

§  3.  What  is  Notation  ?  What  is  Roman  Notation  ?  What 
letters  stand  for  3  ?  for  8  ?  for  10  ?  for  16  ?  for  52  ?  for  70  ? 
&c.,  &c.  What  effect  has  a  letter  of  less  value  when  placed 
before  a  letter  of  greater  value  ?  when  placed  after  ?  What  ef- 
fect has  the  repeating  of  a  letter  ?  What  effect  the  horizontal 
line  over  a  letter  ?  For  what  are  Roman  letters  used  ?  What 
is  the  origin  of  the  character  I?  V?  X?  L?  C?  D?  Show 
this  upon  your  slates.  Write  the  present  year  of  the  Christian 
era  on  your  slates. 

5  4.  Wherein  does  Arabic  differ  from  Roman  notation  ?  Write 
on  the  board  in  backward  order  the  Arabic  characters.  Write  3 
digits  and  2  naughts.  What  do  you  mean  by  a  digit  ?  What 
does  the  word  significant  mean  ? 

26* 


306  APPENDIX.  [CHAP.  n. 

$  5.  How  many  values  have  figures  ?  What  do  they  always 
represent  ?  What  connection  have  their  units  with  their  values  ? 
What  is  meant  by  a  simple  value  ?  Explain  the  local  value  of  a 
figure.  Write  upon  the  board  a  number  of  4  figures,  and  illus- 
trate what  is  meant  by  first  place,  second  place,  &c. ;  first  order, 
second  order,  &c.  Does  "  place"  apply  to  figures,  or  to  the  units 
which  they  represent  ?  To  which  does  "  order"  apply  ?  What 
does  the  first  order  of  units  represent  ?  the  fourth  ?  the  third  ? 
the  second  ?  What  property  pertaining  to  figures  in  a  number  is 
inferred  from  the  illustration  given  ?  What  property  pertains  to 
figures  with  respect  to  the  figure  at  its  left  hand  ? 

\  6.  Illustrate  the  value  of  the  0  in  the  following  number,  three 
thousand  and  three.  Illustrate  the  value  of  the  0  at  the  right  of 
a  number  ;  also,  the  effect  of  cutting  off  a  0  from  the  right  of  a 
number ;  two  00,  three  000.  What  does  the  0  represent,  and 
what  is  its  office  ? 

\  \  7.  8.  How  do  you  write  a  number  that  contains  but  4 
places  ?  How  will  you  write  a  number  containing  three  denom- 
inations of  figures  ?  What  is  the  difference  between  "  places" 
and  "  denominations  ?"  Suggest  a  number  containing  5  denom- 
inations. Express  it  in  figures.  What  is  meant  by  "  periods  ?" 
What  is  the  difference  between  "  periods,"  "  denominations,"  and 
"  places  ?"  In  expressing  numbers  by  figures,  what  particular 
mistake  will  you  be  likely  to  make,  against  which  the  text  cau- 
tions you  ?  Must  the  left-hand  period  always  be  full  ?  Why  ? 
What  is  meant  by  a  period's  being  full  ?  What  is  the  service  of 
the  0  in  notation  ?  Recite  the  first  10  periods,  beginning  with 
units.  Which  system  of  notation  have  you  been  using  ?  How 
does  it  differ  from  the  other  system  ?  and  what  is  the  name  of 
that  other  ? 

\  9.  Wherein  do  Notation  and  Numeration  differ  ?  What  is 
necessary  in  order  to  read  large  numbers  with  facility  ?  Write  a 
number  of  27  figures  on  the  board  and  read  it,  and  explain  its  di- 
visions or  groups.  Give  a  rule  in  your  own  language  for  the 
reading  of  numbers. 


CHAP.  II.]  APPENDIX.  307 

ANSWERS. 

§3.  (1-5.)  XVII;  XLII;  XXVI;  XCV1II ;  CIII. 
(6-11.)  LXXXII;  LVII;  LXXIX  ;  CCCCXXX  ;  DCLXXX  ; 
MMVII.  (12-14.)  CCC;  DCCCCLX ;  M.  (15.)  XLI. 
(16.)  CLXVI.  (17-28.)  Twenty-six ;  one  hundred  and  for- 
ty-four ;  ninety-eight ;  one  thousand  three  hundred  and  twelve  ; 
one  thousand  eight  hundred  and  fifty  ;  one  thousand  nine  hundred 
and  seventy-two ;  five  hundred  thousand ;  five  hundred  and  two 
thousand  seven  hundred  and  seventy ;  one  million ;  one  million 
and  ninety-four ;  one  thousand  six  hundred  and  eighty-eight ; 
one  thousand  seven  hundred  and  seventy-five. 

§  7.  (29-43.)  20  ;  37  ;  98  ;  337  ;  407  ;  2437  ;  6407  ;  8007  ; 
9027;  4006;  3000;  1001;  1100;  101;  1101. 

§8.  (44-49.)  27300;  940200;  36456;  501000;  98000; 
11000.  (50-53.)  46930659  ;  307802509  ;  981000700  ;  10010010. 
(54-57.)  96048073098 ;  807000000006 ;  90000004010 ; 
800006000007.  (58-62.)  48000000000000  ;  609000000000000  ; 
980004000000007;  3000000000002;  9000000002000. 
(63-68.)  36000000000000000000098  ;  4008000000005094 ; 
35000098000000063  ;  900700590  ;  86000000000000005000000- 
000;  900000046000. 

§  9.  (69-75.)  Six  hundred  and  seventy-eight  thousand  two 
hundred  and  ten ;  five  millions,  four  hundred  and  ninety-three 
thousand,  six  hundred  and  seventy-eight ;  four  hundred  and  fifty- 
six  millions,  three  hundred  and  twenty-one  thousand,  nine  hun- 
dred and  eighty ;  seven  hundred  and  seventy-nine  millions,  one 
hundred  and  forty-six  thousand  and  five  ;  forty-two  trillions,  five 
hundred  and  sixty-seven  billions,  one  hundred  and  twenty-three 
thousand,  nine  hundred  and  one  ;  three  hundred  and  twenty-seven 
millions,  nine  hundred  and  eighty  thousand  and  sixty  ;  thirty-two 
quadrillions,  nine  hundred  and  eighty-seven  trillions,  six  hundred 
and  fifty-four  billions,  three  hundred  millions,  and  ninety-eight. 
(76-8O.)  Five  hundred  and  sixty-three  billions,  four  hundred  and 


308  APPENDIX.  [CHAP.  in. 

twenty-eight  millions,  six  hundred  and  seventy  thousand  and  nine  ; 
three  hundred  and  fifty -eight  millions,  nine  hundred  and  twenty- 
thousand,  seven  hundred  and  sixty-one  :  nine  hundred  and  eighty- 
seven  millions,  six  hundred  and  seventy-eight  thousand,  nine 
hundred  and  thircy-two  ;  four  quadrillions,  five  hundred  and  sixty 
trillions,  seven  billions,  nine  hundred  and  eighty  millions,  five 
hundred  and  forty  thousand  and  sixty-eight ;  thirty-three  quintil- 
lions,  four  hundred  and  ninety-two  quadrillions,  six  hundred  and 
seventy-seven  trillions,  five  billions,  three  hundred  and  sixteen 
millions,  eight  hundred  and  ninety-six  thousand,  three  hundred 
and  twenty-one.  (81.)  Twenty  trillions.  (§2.)  Two  hundred 
and  thirty-six  thousand,  eight  hundred  and  forty-seven.  (§3.) 
Thirty-six  millions,  eight  hundred  and  fourteen  thousand,  seven 
hundred  and  twenty-one.  (§4.)  Sixty-eight  millions,  seven  hun- 
dred and  ninety-one  thousand,  seven  hundred  and  fifty-two. 
(85.)  One  hundred  and  forty-four  millions,  nine  hundred  and 
seven  thousand,  six  hundred  and  thirty.  (86.)  Four  hundred 
and  ninety-four  millions,  four  hundred  and  ninety-nine  thousand, 
one  hundred  and  eight.  (87.)  Eight  hundred  and  eighty-three 
thousand,  two  hundred  and  forty-six.  (88.)  Two  millions,  seven 
hundred  and  seventy-four  thousand,  seven  hundred  and  ninety- 
nine.  (89.)  Two  trillions,  four  hundred  and  fifty  billions,  eight 
hundred  and  thirty  millions,  two  hundred  and  forty-one  thousand, 
two  hundred  and  eight.  (9O.)  Three  hundred  and  sixty  quad- 
rillions, seven  hundred  and  eighty-one  trillions,  one  billion,  two 
hundred  and  four  millions,  three  hundred  and  ninety-eight  thou- 
sand, two  hundred  and  ninety-nine.  (91.)  Seventy-three  mil- 
lions, three  hundred  and  seventy-six  thousand,  two  hundred  and 
ninety.  (92.)  Five  hundred  and  sixty-one  millions,  seven  hun- 
dred and  seven  thousand  and  forty-six. 


CHAPTER  III. 

£10.  What  do  you  mean  by  Addition  ?  Add  4  books  to  3 
slates.  What  is  the  one  resulting  number  ?  Reasons  ?  What 
is  this  number  called  ?  Write  an  expression  on  the  board  illus- 


CHAP.  III.]  APPENDIX.  309 

trating  the  sign  plus,  and  the  sign  of  equality.    What  is  meant  by 
numbers  being  of  the  same  kind  or  denomination  ? 

$  11.  In  setting  down  figures  for  addition,  why  must  those  of 
the  same  kind  fall  in  the  same  column  ?  Give  an  example  of  fig- 
ures of  different  kinds.  Write  out  an  example  of  3  numbers  of 
4  figures  each,  where  the  sum  of  each  column  shall  be  less  than 
10  ;  add  and  explain. 

§  12.  Set  down  6  numbers  of  7  figures  each ;  add  and  explain. 
Give  the  rule  for  addition.  Why  must  you  commence  at  the 
right  hand  to  add  ?  On  what  principle  must  you  set  down  the 
right-hand  figure  of  the  sum  under  the  column  added  ?  Why 
carry  the  left-hand  figure  to  the  next  column  ?  What  directions 
does  the  note  give  you  as  to  the  mode  of  adding  ?  What  is  the 
proof  of  addition  ? 

ANSWERS. 

$11.  (1.)  9999.  (2.)  6556.  (3.)  9398.  (4.)  9679. 
(5.)  5659.  (6.)  99968.  f7.)  5539599. 

$  12.  (8.)  125010.  (9.)  177170559.  (1O.)  249770691. 
(11.)  49448176659.  (12.)  23650434530.  (13.)  106014335610. 
(14.)  220957988780.  (15.)  37185329.  (16.)  65285936. 
(17.)  157846611.  (18.)  3308489.  (19.)  5189375. 
(2O.)  5186750.  (21.)  1904798.  (22.)  33577.  (23.)  9364. 
(24.)  365  days.  (25.)  6278  bricks.  (26.)  247031  barrels. 
(27.)  547131  hogsheads.  (28.)  6856886  bales.  (29.)  11119695 
bushels.  (3O.)  30736135  dollars.  (31.)  73376290  pounds. 
(32.)  561707046  pounds.  (33.)  7497567  acres.  (34.)  15028015 
dollars.  (35.)  8108797  dollars.  (36.)  993095817  inhabitants. 
(37.)  50150009  square  miles.  (38.)  70995  individuals. 
(39-42.)  61630939  pounds;  117274121  pounds;  84148377 
pounds;  263053437  pounds.  (43-46.)  17581225  dollars; 
34807174  dollars;  21993877  dollars;  74382276  dollars. 
(47-5O.)  252971606  pounds;  746063693  pounds;  670907293 
pounds;  1669942592  pounds.  (51-54.)  27415498  dollars; 


310  APPENDIX.  [CHAP.  rv. 

68457397  dollars;  46464775  dollars;  142337670  dollars. 
(55-58.)  142337670  dollars;  1669942592  pounds;  74382276 
dollars  ;  263053437  pounds. 


CHAPTER  IV. 

$  $  13,  14,  15,  10.  What  is  meant  by  Subtraction  ?  of  the 
numbers  6  and  8,  one  requiring  to  be  subtracted  from  the  other  ; 
which  is  the  minuend  ?  Why  ?  What  is  the  other  called  ? 
Why  ?  What  is  there  peculiar  in  the  termination  of  these  words  ? 
After  subtracting,  what  is  the  result  called  ?  Why  ?  Write  an 
expression  on  the  board  illustrating  the  symbols  plus,  minus,  and 
equality.  Write  an  example,  r.herein  each  figure  of  the  subtra- 
hend shall  be  less  than  its  corresponding  figure  of  the  minuend  ; 
subtract  and  explain.  Write  an  example,  wherein  figures  of  the 
subtrahend  are  larger  than  corresponding  figures  of  the  minuend ; 
subtract  and  explain.  Give  the  rule  and  the  reason  for  every 
statement  in  it.  How  do  you  prove  your  work  in  subtraction  ? 

ANSWERS. 

$13.  (1-12.)  2;  9;  4;  6;  10;  17;  5;  8;  7;  11;  16; 
14.  (13-25.)  2;  4;  6;  16;  14;  13;  10  ;  12;  15 ;  7 ;  3  ;  5 ; 
8.  (26-39.)  2  ;  5  ;  8  ;  11  ;  14  ;  3  ;  6  ;  9  ;  12  ;  15  ;  4  ;-  7  ; 
10;  13.  (40-52.)  2  ;  5  ;  8  ;  11  ;  14  ;  3  ;  6  ;  9  ;  12  ;  4  ;  7  ; 
10;  13.  (53-64.)  2  ;  5  ;  8  ;  11  ;  3  ;  6;  9  ;  12  ;  4  ;  7  ;  10  ; 
13.  (65-75.)  2;  5;  8;  11;  3;  6;  9;  12;  4;  7;  10. 
(76-85.)  2  ;  5  ;  8  ;  11  ;  3  ;  6  ;  9  ;  4  ;  1  ;  10. 

§14.  (§6.)  201.  (§7.)  2211.  (8§.)  10154.  (89.)  150110. 
(9O.)  166304310. 

1 15.  (91.)  1093.  (92.)  3328.  (93.)  67467.  (94.)  25485. 
(95.)  1089088.  (96.)  20891.  (97.)  669042.  (98.)  9443544813. 
(99.)  6066069034.  (1OO.)  1075415.  (1O1.)  51113291. 
(102.)  1357322792.  (1O3.)  6889336062.  (1O4.)  8849208. 
(105.)  969116902.  (1O6.)  8365421086.  (1O7.)  4219238873. 
C1O8.)  7023226.  (1O9.)  999998.  (11O.)  364635. 


CHAP.  V.]  APPENDIX.  311 

(111.)  352  years.  (112.)  2142  dollars.  (113.)  1464398 
bushels.  (114.)  201 324  barrels.  (115.)  26  miles.  (116.)  67 
years.  (117.)  5181  votes.  (118.)  2210  votes.  (119.)  5333865 
dollars.  (12O.)  105588748  pieces.  (121.)  Gold  exceeded 
silver  by  1475597  dollars ;  gold  exceeded  copper  by  3992969 
dollars ;  silver  exceeded  copper  by  2517372  dollars.  (122.)  3413 
post-offices;  31166  miles  of  road.  (123.)  515915  inhabitants. 
(124.)  147  dollars.  (125.)  564  dollars.  (126.)  1277  dol- 
lars. (127.)  225  miles.  (128.)  168  dollars.  (129.)  In 
1840  total  value  was  3426632  dollars  ;  in  1841,  2240320  dollars  ; 
1840  exceeded  1841  by  1186312  dollars  ;  in  1840,  silver  exceed- 
ed gold  by  51401  dollars,  silver  exceeded  copper  by  1702076  dol- 
lars ;  in  1841,  silver  exceeded  gold  by  41153  dollars,  silver  ex- 
ceeded copper  by  1116777  dollars.  (13O.)  20500  dollars. 
(131.)  1000000  total  volumes  ;  96000  total  manuscripts  ;  400000 
excess  of  volumes  in  Paris  library  above  those  in  Vienna  library ; 
64000  excess  of  manuscripts ;  904000  total  excess  of  volumes 
above  manuscripts.  (132.)  975  total  number  of  votes;  113 
number  of  votes  in  majority. 

CHAPTER  V. 

§  17.  What  is  Multiplication  ?  What  is  the  difference  be- 
tween the  multiplier  and  the  multiplicand  ?  What  are  they  called  ? 
Why  ?  What  is  the  difference  between  a  factor  and  a  product  ? 
Write  an  example  by  means  of  symbols,  and  show  which  is  mul- 
tiplier, which  is  multiplicand,  which  are  the  factors,  which  is  the 
product.  Suppose  multiplier  and  multiplicand  change  places, 
what  is  the  result  ?  What  is  a  square  ?  What  is  a  square  root  ? 
Illustrate. 

§  §  18,  19.  Perform  an  example,  with  a  multiplier  of  one  fig- 
ure, and  explain  the  process.  Perform  an  example  with  a  multi- 
plier of  3  figures.  Suppose  there  is  a  0  in  the  multiplier,  how 
do  you  proceed  ?  Of  what  denomination  is  the  product,  if  you 
multiply  together  units  and  units  ?  units  and  hundreds  ?  tens  and 
hundreds  ?  tens  and  tens  ?  hundreds  and  thousands  ?  tens  and 


312  APPENDIX.  [CHAP.  -v. 

ten-thousands  ?  How  does  the  denomination  of  a  product  guide 
you  as  to  the  place  which  the  first  figure  of  any  partial  product 
should  occupy  ? 

§  $  20,  21.  What  is  another  definition  of  multiplication  ?  Show 
how  this  is  true.  What  is  necessary  with  regard  to  the  numbers 
added  that  an  exercise  in  addition  might  be  turned  into  an  exer- 
cise in  multiplication  ?  What  is  the  rule  for  multiplication  ? 

§  22.  What  is  the  method  of  proof  ? 

$  23.  Sometimes  one  or  both  factors  will  have  O's  at  the  right ; 
what  must  be  done  in  such  case  ?  Why  ? 

§  24.  What  do  you  understand  by  a  composite  number  ?  Give 
an  instance  of  a  number  that  is  composite,  and  of  one  that  is  not. 
If  a  multiplier  be  composite,  how  may  you  proceed  ?  .  Give  an 
example.  Give  the  rule  in  your  own  words. 

ANSWERS. 

§  17.  (1-11.)  4  ;  9  ;  16  ;  25  ;  36  ;  49  ;  64  ;  81  ;  100  ;  121  ; 
144.  (12-22.)  5;  7;  2;  3;  4;  6;  8;  9;  10;  12;  11. 
(23-27.)  32;  18;  84;  64;  81.  (28-36.)  12;  18;  24;  30 
36  ;  9  ;  27  ;  21  ;  15.  (37-47.)  8  ;  16  ;  24  ;  32  ;  40  ;  48  ;  12 
20  ;  28  ;  36  ;  44.  (48-58.)  15  ;  30  ;  45  ;  60  ;  10  ;  25  ;  40 
55;  20;  35;  50.  (59-69.)  18  ;  36 ;  54 ;  72  ;  12;  30;  48 
66;  24;  42;  60.  (7O-§O.)  21;  42;  63;  84*;  14;  35;  56 
77  ;  28  ;  49 ;  70.  (81-135.)  24  ;  48  ;  72  ;  96  ;  16  ;  40  ;  64 
88  ;  32  ;  56  ;  80  ;  27  ;  54  ;  81  ;  108  ;  18  ;  45  ;  72  ;  99  ;  36 
63;  90;  30;  60;  90;  120;  20;  50;  80;  110;  40;  70;  50 
33  ;  66;  99  ;  132  ;  22  ;  55  ;  88  ;  121 ;  44  ;  77  ;  110  ;  36  ;  72 
108  ;  144  ;  24  ;  60  ;  96  ;  132  ;  48  ;  84  ;  120.  (136-142.)  30 
36;  14;  15;  50;  121  ;  56. 

§18.  (143.)  2468.  (144.)  468312.  (145.)  2449512. 
(146.)  4488270.  (147.)  6020736.  (148.)  1439694746. 
(149.)  52248187648.  (15O.)  8019276804702.  (151.) 
44025632.  (152-16O.)  62972;  94458;  125944;  157430; 
188916;  220402;  251888  ;  283374 ;  1385384.  (161-169.) 


OHAP.  V.]  APPENDIX.  313 

80785809  ;  71809608  ;  62833407  ;  53857206  ;  44881005  ; 
35904804  ;  26928603  ;  17952402;  394952844.  (17O-172.) 
5216648;  1956243;  3260405.  (178-180.)  61232;  7849350; 
51744;  81195867;  17295;  10172519;  49864787;  149212794. 

$  22.  (1§1.)  1542382864.  (182.)  55056418756.  (183.) 
9472469137.  (184.)  3937919100705.  (185.)  28574677132. 
(186.)  23070596606.  (1ST.)  254087145206.  (188.) 
1270996912224.  (189-195.)  529259254443  ;  12138394951269  ; 
8141111037027;  108215060743638;  949354188891741; 
121932631112635269;  123011009127513387.  (196.)  1665- 
3188645286.  (197-198.)  17494334544;  57482188.  (199- 
2OO.)  33793364;  837852. 

§  23.  (2O1-2O4.)  764290 ;  7642900 ;  76429000  ;  764290000. 
(2O5-2O8.)  1974800;  19748000;  197480000;  1974800000. 
(2O9-212.)  19626000;  196260000;  1962600000;  19626000000. 
(213-216.)  32001280000;  320012800000;  3200128000000; 
32001280000000.  (217-218.)  1161253800000000; 
28755071047000000. 

§24.  (219.)  10220.  (220-224.)  8976;  6732;  13464; 
23562  ;  40392.  (225-23O.)  82960332  ;  34566805  ;  79997463  ; 
76046971;  63207872;  39504920.  (231-237.)  119376; 
17409;  82071;  24870;  12435;  19896;  9948.  (238.)  18750 
bricks.  (239.)  2033  bushels.  (24O.)  629  miles.  (241.) 
53200  pounds.  (242.)  363  dollars.  (243.)  480  dollars. 
(244.)  8361574  dollars.  (245.)  2756  bushels.  (246.)  75798 
dollars.  (247.)  2715  dollars.  (248.)  16199568  hours. 
(249.)  3760128  cubic  inches.  (25O.)  141440  dollars.  (251.) 
7560  miles.  (252.)  59568000  miles.  (253.)  479544  dollars. 
(254.)  2205  dollars.  (255.)  109  miles.  (256.)  215  dollars. 
(257.)  32  acres  at  198  dollars.  (258.)  149  miles.  (259.)  Lose 
27  dollars.  (26O.)  941618440000.  (261.)  295  dollars. 
(262.)  14553  cubic  inches. 

27 


314  APPENDIX.  [CHAP.  vi. 


CHAPTER  VI. 

{ $  25,  26.  What  is  Division  ?  In  the  example  6-1-2,  what  is 
the  6  called  ?  What  the  2  ?  Why  ?  What  is  the  result  called  ? 
What  is  the  remainder  ?  Write  an  example  containing  the  sym- 
bol of  division,  and  the  symbol  of  equality.  How  is  an  accurate 
quotient  sometimes  to  be  expressed  ?  Under  what  circumstances 
must  it  be  so  expressed  ?  Give  an  example  of  your  own  upon 
the  board,  of  the  division  of  a  number  by  a  single  digit.  Explain 
as  you  go  along.  Show,  by  an  example,  the  reason  for  the  state- 
ment that  division  is  a  concise  way  of  performing  several  subtrac- 
tions. 

§  27.  What  is  the  difference  between  short  division  and  long 
division  ?  What  is  the  rule  for  short  division  ?  What  for  long 
division  ?  Perform  an  example  in  each,  and  apply  the  rule  step 
by  step  as  you  proceed.  Illustrate  the  notes  after  rule  for  long 
division. 

$  §  28,  29.  How  do  you  prove  your  work  in  division  ?  How 
do  you  proceed  when  your  divisor  ends  with  one  or  more  naughts  ? 
Suppose  a  digit  be  cut  off  from  the  right  of  a  number,  what  effect 
has  it  ?  What  does  the  digrit  so  cut  off  represent  ?  How  is  the 
true  remainder  found  after  dividing  by  a  divisor  with  naughts  cut 
off?  If  there  be  a  remainder  after  such  division,  is  it  of  the  same 
denomination  as  the  digit  or  digits  cut  off?  Illustrate  by  an  ex- 
ample on  the  blackboard  the  division  of  a  number  by  a  composite 
divisor.  Show  how  you  find  the  true  remainder,  and  give  the 
reason  for  each  step. 

ANSWERS. 

§  25.    (1-12.)    1  ;  3  ;  4  ;  6  ;  2  ;  5  ;  7  ;  9  ;  8  ;  10  ;  12  ;  1 1. 

(13-24.)  1  ;  3;  5;  7;  9;  11  ;  12;  2;  4;  6;  8;  10 
(25-34.)  2;  4;  6;  8;  10;  12;  3;.  5;  7;  9.  (35-45.)  2 
4;  6:  8;  10;  12;  3;  5;  7;  9;  11.  (46-54.)  2;  4;  6;  8 
10  ;  3  ;  5  ;  7  ;  9.  (55-65.)  2  ;  4  ;  6  ;  8  ;  10  ;  12  ;  3  ;  5  ;  7 
9;  11.  (66-,76f)  2;  4:6;  8:  10:  12:  3;  5:  7:  9;  11 


CHAP.  VI.] 


APPENDIX. 


315 


(77-87.)  2;  4;  6;  8;  3;  5;  7;  9;  11;  10;  12.  (88-1OO.) 
2;  4;  6;  9;  8;  7;  3;  5;  10;  13;  12;  11  ;  14.  (1O1-1O9.) 
2;  4;  6;  3;  7;  10;  8;  11;  12.  (11O-12O.)  2;  4;  5;  8; 
7  ;  12  ;  10  ;  11  ;  3  ;  6  ;  9.  (121-129.)  9;4;7;7;20;7; 
12;  5;  9.  '(13O-134.)  3  ;  4  ;  4  ;  4  ;  4. 

$26.  (135.)  12301.  (136.)  1682227f.  (137.)  1786213£. 
(138.)  5315728|.  (139.)  128236331^.  (14O.)  115134608|, 


(141.)  26063098 


(142.)  8491229. 


(143.)  95665602. 


.          .       .        .       .        . 

(144.)  127160495^.  (145.)  1315020576.  (146.)  1357802469. 
(147-154.)  1739451;  115963|;  86972£  ;  69578};  5798lf  ; 
49698^  ;  43486|  ;  38654f.  (155-162.)  38270653^;  25513769  ; 
191353261;  15308261f;  12756884f;  10934472^;  9567663f  ; 
8504589|.  (163-1TO.)  44882U;  2992141  ;  224410|;  179528^; 
149607};  128234f  ;  112205J;  9"9738i.  (171-17§.)  3826954: 
2551302f  ;  1913477;  1530781|;  1275651f  ;  1093415|  ;  956738|  ; 
850434|.   (179-183.)  90536f  ;  6646090^;  1327581667; 
7815886f  ;  1666481  j.   (184-213.)  181072§;  13292180^; 
26551623334;  15631772f;  33329621; 
11723829f;  2499721^ 
6699331J;  1428412^; 
5861914§;   1249860|; 


19913717500f 
113792671431 

9956858750|; 
8850541 11  If; 


5210590|; 


1110987^; 


15930974000f  ;  9379063|;  1999777|. 


135804f 

77602^ 

67902| 

60357| ; 
108643^;  7975308}; 


9969135J ; 
5696648f; 
49845671 ; 
4430726^ ; 


28.  (214-218.)  34424|f; 

?p   (219-224.)   3437681^;  210948|f;  947ll 


122128l£; 

894143£|f ;  1162136^; 


(236-243.) 


2882566f Jf§ ; 
15927811^.; 
(244-250.) 


(225-235.) 
;  2058970|-Jf  ; 
345016  J}}|;  276036^%- 


98HHi 
70056.; 


Hf 


2746^ 

(251-257.)     156472f|i;  .   8J?|J}J; 
12890625;  109376;  681j4/AV 


234 


316  APPENDIX.  [CHAP.  vir. 

§29.  (25§-263.)  864^;  18296;  493827^  ;  17283945^. 
6789734;  18982660^,  (264-271.)  289763412^;  231810- 
729f£|;  144881706//o;  99347455*j&;  100787273£{$;  17385- 
804f$gJ;  869290&J}};  77270ft  JgjfJ.  (272-273.)  16- 
IfffM;  28860 AVA-  (274-275.)  10037^%; 
(276-270.)  29123  j'AV;  90jftW;  10}?Ji£& 
(280-285.)  11960f|jfJ;  25647lf^ ;  22103fff™;  95- 
9A7o;  95S}H§fjf.  (286-289.)  2120T3,%VT; 
918412|f^;  144271i§^.  (29O-295.)  234,  r.  27  ; 
2245,  r.  3;  133,  r.  15;  221,  r.  30;  11438,  r.  7;  15677,  r.  3. 
(296.)  1994  dollars.  (297.)  2776  sheep.  (298.)  991  dollars. 
(299.)  974  dollars.  (3OO.)  1177  dollars.  (3O1.)  210  acres. 
(3O2.)  1st  56,  2d  70,  3d  105,  4th  105.  (303.)  412  dollars. 
(3O4.)  249  acres.  (3O5.)  11875000  miles.  (306.)  4545£f 
cubic  feet.  (3O7.)  125  days.  (3O8.)  20T2/37g(V4g-  dollars! 
(3O9.)  103368000  hours,  4307000  days,  11800  years.  (31O.) 
856  barrels,  107  trees.  (311.)  Each  had  900  dollars.  (312.) 
2191  dollars.  (313.)  1632000  miles  in  one  day,  595680000 
miles  in  one  year.  (314.)  3100  dollars.  (315.)  2  days. 
(316.)  They  will  meet  in  5  hours,  at  a  distance  of  75  miles. 
(317.>54  dollars.  (318.)  125  dollars.  (319.)  2  dollars. 
(32O.)  49  miles. 

CHAPTER  VII. 

§  3O.  What  do  you  understand  by  a  problem  ?  by  a  principle  ? 
Show  how  problem  a  is  founded  upon  the  preceding  rules. 
Illustrate  problem  b.  Illustrate  problem  c.  Illustrate  problem  d. 
Illustrate  problem  e ;  problem  jf;  problem  g.  Can  you  give  a 
practical  example  (not  taken  from  the  book)  of  the  use  of  any  one 
of  the  preceding  problems  ? 

5  31.  What  are  the  names  of  the  quantities  used  in  division  ? 
What  effect  has  the  multiplication  of  a  divisor  upon  the  result  in 
division  ?  What  the  division  of  a  divisor  ?  What  the  multipli- 
cation or  division  of  a  dividend  ?  How  is  the  remainder  affected 


CHAP.  VIII.]  APPENDIX.  317 

by  such  operations  upon  divisor  or  dividend  ?  What  relation  has 
the  quotient  to  the  divisor  ?  to  the  dividend  ?  to  the  remainder  ? 
If  the  remainder  be  as  large  as  the  divisor,  what  is  to  be  done  ? 
Can  the  remainder  ever  be  as  large  as  the  quotient  ?  Can  it  ever 
oe  exactly  equal  to  the  quotient  ?  Give  examples.  Illustrate 
each  principle  in  the  section  ;  in  #,  b,  c,  d,  e,f,  g. 

ANSWERS. 

§30.  (I.)  123423434.  (2.)  343148.  (3.)  59831.  (4.) 
879465.  (5.)  1037654321,771350011.  (6.)  23474.  (7.)  4567031. 
f§.)  34678  dollars,  13787  dollars.  (D.)  1240578.  (IO.)  354. 
(II.)  1521808704.  (12.)  4556  votes,  4181  votes.  (13.) 
144000000  miles,  95000000  miles.  (14.)  49  trees.  (15.)  35405 
dollars.  (16.)  5718  dollars.  (17.)  45441  hills.  (1§.)  42  gal- 
lons, 23  gallons.  (19.)  11  miles,  7  miles.  (2O.)  646  dollars. 
(21.)  44  years  old.  (22.)  5  dollars.  (gJI.)  157  barrels. 
(24.)  101  cubic  feet.  (25.)  1728  cubic  inches. 


CHAPTER  VIII. 

\  §  32,  33.  What  is  the  difference  between  a  prime  and  a 
composite  number  ?  Give  examples.  To  what  extent  can  you 
determine  upon  inspection  whether  a  number  is  prime  or  not  ? 
What  is  an  even  number  ?  an  odd  ?  Show  how  and  why  it  is 
that  a  number,  the  sum  of  whose  digits  is  equal  to  9,  is  itself  di- 
visible by  9.  Show  how  and  why  the  same  thing  is  true  of  3. 

\  34.  What  is  a  divisor  ?  a  common  divisor  ?  the  greatest 
common  divisor  ?  Show  this,  by  analyzing  numbers.  Can  prime 
numbers  have  common  divisors  ?  Give  reason.  What  is  the 
common  divisor  of  two  numbers  that  are  prime  to  each  other  ? 

\  35.  How  is  the  greatest  common  divisor  found  by  the  process 
of  long  division  ?  Explain  this  process,  giving  the  reasons  for 
each  step  as  they  ara  explained  in  (a)  or  (b).  How  would  you 
proceed  to  find  the  greatest  common  divisor  of  three  or  four  num- 
bers? 

27* 


318  APPENDIX.  [CHAP,  vm 

§  36.  What  is  a  multiple  ?  a  common  multiple  ?  the  least  com- 
mon multiple  ?  What  is  the  difference  between  the  least  common 
multiple  and  the  greatest  common  divisor  ?  Is  the  greatest  com- 
mon divisor  of  two  or  more  numbers  a  factor  of  their  least  com- 
mon multiple  ?  If  so,  show  how.  What  would  be  the  other  fac- 
tor of  such  multiple  ?  How  many  multiples  may  any  number 
have  ?  Show  how  to  find  the  least  common  multiple  by  decom- 
posing into  primes. 

§  37.  Show  the  same  by  the  process  under  present  section. 
Explain  how  this  process  agrees  with  the  former. 

$  38.  What  is  cancelation  ?  Is  it  employed  in  subtraction  or 
in  addition  ?  Give  an  example  of  cancelation.  In  what  way  is 
it  useful  ? 

ANSWERS. 

533.  (1-8.)  2X2X3;  2X7  ;  3x5  ;  2x2X2X2  ;  2X3X3; 
2X2X5;  2X11;  2X2X2X3.  (9-16.)  5x5;  2X13;  3X3 
X3;  2X2X7;  2X3x5;  2X2X2X2X2;  3X11;  2X17. 
(17-24.)  5X7;  2X2X3X3;  2X19;  3X13;  2X2X2X5; 
2X3X7;2X2X11;  3X3X5.  (25-32.)  2X23;  2X2X2X2X3; 
7X7;  2X5X5;  3X17;  2X2X13;  2X3X3X3;  5X11. 
(33-11.)  2X2X2X7;  3X19;  2X29;  2X2X3X5;  2X31; 
3X3X7;  2X2X2X2X2X2;  5X13;  2X3X11.  (42-5O.) 
2X2X17;  3X23;  2X5X7;  2X2X2X3X3;  5X17;  3X29; 
2X3X3X5;  2X2X2X2X2X3  ;  2X7X7.  (51-57.)  2X3X17; 
3X37;  7X17;  5X5X5;  2X3X23;  2X73;  5X31.  (5§-63.) 
2X7X11;  2X83;  2X89;  11X19;  2X3X3X13;  7X37. 
(64-89.)  3X103;  2X3X61  ;  3X5X5X5;  2X2X101;  11X43; 
2X2X131.  (70-76.)  2X2X2X11X13;  2X2X2x2x5x13; 
2X2X2X3X3X19;  2X2X2X3X3X17;  2X2X2X823;  3X3 
X11X797;  2X2X3X5179.  (77-83.)  2X3X7X13X19; 
2X2X5X43X89;  2X131X241;  2X2X3X5X5X263;  2X2X 
2X2X2X3X67;  3X17X1913;  2X3X14951.  (§4-9O.)  2X23 
X163;  2X31X907;  2X5X5X5X199;  3X1111;  3X7X4759; 
3X5X3251:  2X5x7x1327. 


CHAP.  IX.]  APPENDIX.  319 

§  34.  (91-94.)  12  ;  they  have  none  ;  45  ;  5.  (95-99.)  22  ; 
4;  8;  8;  66.  (1OO-1O3.)  6;  6;  18;  18.  (1O4-1O6.)  2; 
6;  14.  (1O7-1O9.)  2 ;  2  ;  2. 

§35.  (110-117.)  They  have  none;  they  have  none;  they 
have  none;  5;  10;  12;  16;  234.  (118-121.)  161;  203; 
35;  111.  (122-124.)  3;  3;  406.  (125-127.)  They 
have  none;  they  have  none;  they  have  none.  (128-13O.) 
They  have  none ;  they  have  none ;  101.  (131-132.)  203  ;  555. 

^37.  (133-13§.)  48;  120;  616;  1517;  360;  24.  (139- 
143.)  315;  2520;  1008;  27720  ;  720.  (144-147.)  360  ; 
100;  1620;  920.  (148-151.)  840;  210;  7106;  128700. 
(152-156.)  39000  ;  336600  ;  510510  ;  4560  ;  3360. 

§38.    (157.)     14.      (158.)     119.      (159-16O.)     12;    8. 

(161-162.)  48;  24.  (163-169.)  576;  432;  288;  216; 
144;  108;  72.  (17O-172.)  20;  90;  40.  (173-175.) 
16200;  21900;  59130.  (176-179.)  21;  121;  363;  847. 
(18O-183.)  3276;  378;  416;  288.  (184-187.)  55;  88; 
40;  66.  (188-190.)  300;  75;  80. 

CHAPTER  IX. 

§  39.  What  is  a  fraction  ?  What  does  the  word  mean  ?  In 
how  many  ways  may  a  given  fraction  be  represented  ?  In  how 
many  ways  may  a  fraction  in  the  common  form  be  read  ?  What 
is  the  name  of  the  term  above  the  line  ?  and  what  does  it  denote  ? 
What  the  term  below  the  line  ?  Express  by  a  fraction  the  value 
of  unity.  Show  the  difference  between  a  proper  and  an  improper 
fraction.  Write  a  mixed  number.  What  is  meant  by  an  inte- 
ger ?  Write  a  compound  fraction.  What  is  the  difference  be- 
tween a  compound  and  a  complex  fraction  ?  When  is  a  fraction 
inverted  ?  Write  the  nine  digits  as  improper  fractions.  What 
two  kinds  of  fractions  are  spoken  of  ? 

5  40.  Upon  what  are  common  fractions  founded  ?  Illustrate. 
What,  then,  does  a  fraction  express  ?  What  connection  has  §  31 


320  APPENDIX.  [CHAI .  IX. 

with  the  subject,  of  fractions  ?  What  propositions  are  deduced 
from  that  section  ?  Illustrate  each  proposition  by  reference  to 
the  principle  in  division  on  which  it  is  founded. 

§  41.  Define  reduction.  Illustrate  it.  What  is  meant  by 
lower  terms  ?  by  lowest  terms  ?  What  have  these  expressions  to 
do  with  reduction  ?  What  has  the  greatest  common  divisor  to  do 
with  reduction  to  lowest  terms  ? 

§  42.  Define  an  improper  fraction,  a  whole  number,  a  mixed 
number.  How  do  you  reduce  the  first  to  the  second  or  the  third  ? 
Give  the  rule. 

\  43.  Illustrate  the  reduction  of  a  whole  or  mixed  number  to 
an  improper  fraction.  Give  rule. 

§  44.  Illustrate  the  reduction  of  compound  fractions  to  simple 
ones.  Give  rule.  In  multiplying  a  fraction  by  a  whole  number, 
what  form  may  the  whole  number  take  ? 

}  45.  Define  the  term  common  denominator.  How  is  this 
found  ?  What  is  the  change  produced  upon  fractions  by  this 
process  ?  Give  the  rule.  Give  an  example,  on  the  board,  of  the 
reduction  of  a  mixed  number  and  a  compound  fraction  to  a  com- 
mon denominator. 

\  46.  How  does  the  least  common  denominator  differ  from  the 
common  denominator  ?  Of  what  service  is  the  least  common 
multiple  in  this  connection  ?  Give  the  rule. 

§  \  47-50.  What  is  necessary  before  fractions  can  be  added  ? 
Rule.  What  is  necessary  before  fractions  can  be  subtracted  ? 
Rule.  Give  the  rule  for  the  multiplication  of  fractions.  Explain 
and  illustrate  each  step.  Illustrate  the  two  methods  of  division  of 
fractions.  What  is  the  principle  involved  in  the  first  method  ? 
What  principles  are  involved  in  the  second  method  ?  Give  rv^e. 
Is  this  inversion  of  the  terms  of  the  divisor  a  mechanical  he*  - 
does  it  involve  a  principle  ? 


CHAP.  IX.]  APPENDIX.  321 

§  51.  What  is  a  reciprocal  ?  of  an  integer  ?  of  a  fraction  ? 
How  may  an  operation  in  division  be  included  under  that  of  mul- 
tiplication ?  Illustrate. 

ANSWERS. 

§41.  (1-5.)  f;  J;  J;  i;  \.     (6-11.)   f  ;  |;  i;  J;  f;  }; 

(12-16.)  l;  Ifff;  flf;  &V;  f.  (1T-2O.)  TIT;  f-'fj; 
jllli;  f.  (21-25.)  iff;  f£;  |6;  1;  3ff  (26-3O.)  if; 
AVs  T7A;  T9oV;  it-  (31-40.)  f§;  ffl;  |§;  ffj;  A'Ai  IWi 

820  .  441  .   971  .   971 
T43T>  3S3>  2SS2>  2a$2' 


§  42.  (41-5O.)  3;  3;  12;  12;  48;  46;  8;  12;  1000; 
11110.  (51-64.)  2f;  11?;  7J;  9^;  1J;  18f;  10T5r;  10J4; 
699?;  8lf|f  ;  34^;  32TVT;  39f|;  lOjfJf.  (65-68.) 
6;  203f  ;  3. 

c  XI  O      /rfJQ    •yi^      36.      45.      54.      63.      72.      81.    90 

$  W.    (t>^-7».;    -4-,    -5-,    -tf~m,    -if-,    ~I->    ~»-j    To^- 

ta«»N9.  10.  37.  25.  15.  75.  64.  50.  295.  1183. 
"OV  2»  "3~»  '5->  T-J  1T»  ~U"»  ~J"  »  T3~J  -T2->  ~32  '» 
3  "4  5  /'ft*?  O<F>  ^  29817.  1554.  1314.  ISO  .  30376.  38283. 

iW-     (87-»».^  -3^5-5  -41-  »   53T  >  ~T¥->    -«r~  >    -3-y  > 

_W_;    104364  .    7_1_0^_0  .    38113.    (97_99.)    164.5J  .     1J.09J  . 


5  45.  (122-129.)  },  f  ;  tf,  #  ;  J},  ||  ;  fj,  fj  ;  |f  J,  Jfi  ; 

Hi.  iff  ;  if*.  Ill  J  UJ,  ill-  (iso-ise.)  Tv7,  AV,  VA  ;  if, 

8         6    .     40     45      48   .     360      378      384.     1188      1200     1210.    J2520     J35_35 
•o4">  ^4   J    tfO'  ¥0»  70"  '    ¥32"»  4~S2>   4"J2"  '    T3^2F'  T3"2"(F'   1  3  2^  '    273"D'    27T(7' 

Iff!  ;  «J,  ?W,  «J-  (137-139.)  A,  f|,  ft  ;  JJ,  ^3%°-,  -V^6-  ; 


(100-101.)  |;  f.  (102-109.)  fa  T3/,;  };  J; 

f;  A;  W-=Mi-  (no-iie.)  A;  A;  A;  -W-= 

;  i;  I;  orV  (HY-121.)  J}};  ^3;  Yfea=3JU;  jff 


^  46.   (140-146.)  T«L,  T<L  T\  ;  *|,  Jf,  |§  ;  *,*  ;  If,  H  ;  If, 

45.     195      234     240.     100      144     135  (I/W     t  ftS  ^       5-°       -5-fi       _5  _S     . 

If  J    26"Ti'  26"o'  "2¥o  J    T¥o>  T^O'  T'So-'        V-11  *«-••.«>«*•;    JOQ-,   12<T'   12^  ' 

1.5         18         56.       10<i          130        3R.1RO         S4        117.        88         1502025. 
T2(F'  T20'  T2~(»"  '    ~3^"~»    ~3"iT~5    Uff  '    T^T-'    TUTT'    Tlio'     3":1T>'    3"37»  ?    55fl    > 


322  APPENDIX.  [CHAP,  ix 

30  40   195   1  i  .   21    70    30     40".  40   45   48   SO   27.   80 

S(T>  FOJ  ~Tff->  -60  >  2Ti>»  ^nr>  ^nr»   2105  ST>  ¥0?  FOJ  *T>  tftfj  ITUJ 

30         72       100108         7  /I  ^ft  "\      1260         840  630         fi  0  4         420 

T20>  T2~ff>  T2~in  T20"»  T2ff'        l***»y     2~S20'     252tf'     252TT'  ^320"'   2S2ff» 


$47.  (157-162.)  |=ii  ;  A;  A.^33.=2,3.  _7_V.  f|=j 
=2J.  (163-168.)  21;  25%  ;  2|J};  SjifrJ  fj;  1&.  (169- 
174.)  7TV;  85V;  fi;  lf!J;  W!i;  l-r4A 


rV;  3011;  18H; 


(187-195.)  A;  *;  H;  iJ;  *fT;  If;  A;  A;  ii- 
(196-203.)  A;  j;  !;  A  ;  i;  sVno  ;  AV;  «-  (204-209.) 

i  ;  if  ;  39^  ;  1  A  ;  r3A  ;  ^i-    (210-213.)  4  ;  A  ;  6A  ;  AV 

(214-218.)  ff;  /A;  //^  .   _47«_  .    _2^     (219-222.) 


549.  (223-233.)    J;    J;  A;    ^5  i;  ?5  A?  *5  T\>  K; 
Jj.      (234-240.)    ASriir;    !  5«;«  ;  T°A  J   oV      (241- 

243.)  7%;T4A;^V  (244-246.)  1TL;  15;  2|.  (247- 
249.)  J;  f;73|.  (25O-254.)  A;  If;  5J  ;  26J  ;  82J. 
(255.)  1T¥A-  (256.)  ,V  (257.)  ^.  (258.)  127?J}. 


550.  (259-269.)  2;  1J;  1|;  1J;  1T'T;  lA;  1A  S  I;  Ai 

If!   if-     (270-274.)    2j;  1^-;   l;   |6.  ^     (2T5_278.) 

AV;  A;  sf;  iff.     (279.)  i§.     (280.)  $#.    (281.) 

4jff*fi-     (282.)  3|.     (283-287.)    1^5  Ifff  5  8-^;  T^; 


5  51.  (288-295.)  |  ;  i  ;  J  ;  A  J  A  5  A  5  A  5  rh- 
303.)  |;  |;  |;  f  ;  };  «;  |;  JJL.     (3O4-3O9.)  f  ;    1;    T\; 
37ff5  f  ;  yVV     (310-313.)  31;  1J4-;  l^j  3|.     (314-318.) 

i-3^;  iJ;  A;  T9^;  H-    (319-324.)  };  f  ;  fff};  tf;  |*j; 

A-     (325-329.)   lT|-f  ;  7f  ;  3T8T5  2|fl  ;  1^.     (33O-334.) 

1  1  W-;  ft;  Ht;  ^T1-   (335-339.)  j;  •;  /F;  237;  200. 


OHAP.  X.J  APPENDIX.  323 

(340-344.)  T<i,  A.  A;  K.  K.  4*»  *§»  W;  K»  f*.  tf.  A; 
AV*  A%»  iVA,  A°A;  SJJi,  fttf.  i«i»  MM-  (345-346.) 

1A;2«.  (347.)  f  (348-350.)  £;  T-JT  ;  ^.  (351.)  30 
feet.  (352.)  A.'s  844,  B.'s  633.  (353.)  A.'s  1402  dollars, 
B.'s  2804  dollars.  (354.)  A.'s  claim  6000  dollars,  B.'s  5000 
dollars,  C.'s  5000  dollars.  (355.)  A.  must  have  1,  B.  J.  (356.) 
1230  sheep.  (357.)  A.  had  154  dollars,  B.  165,  C.  264,  and  D. 
33.  (358.)  J.  (359.)  12000  dollars.  (36O.)  She  gave  away 
in  all  9500  dollars.  To  the  Fire  Department  Fund  3000  dollars  ; 
to  the  Musical  Fund  Society  2000  dollars  ;  and  to  each  of  the 
other  societies  500  dollars.  (361.)  Iff  baskets.  (362-363.) 
/T  dollars;  23^-  cents.  (364.)  19£J  barrels.  (365.)  ljv  bush- 
els, ||  of  a  bushel,  1^  bushels,  T3^  of  a  bushel.  (366.)  A.  went 
52  miles,  B.  43  miles.  (367.)  100  feet.  (36§.)  4000  dollars. 
(369.)  £,  jh,  TV\,  AVVv  (3*°-)  A.'s  23  dollars,  B.'s  23, 
C.'s  21,  andD.'s21. 


CHAPTER  X. 

§  52.  What  is  the  difference  between  a  common  and  a  Decimal 
fraction  ?  What  gives  the  name  decimal  to  the  fraction  ?  Which 
is  the  most,  3  units  or  4  hundredths  ?  Wherein  does  a  decimal 
require  a  different  treatment  from  a  whole  number  ?  What  is  the 
use  of  the  point  ?  What  are  the  names  of  the  first  six  decimal 
places,  beginning  at  the  left  hand  ?  What  is  the  difference  in 
value  between  1  ten  and  1  tenth  ?  between  1  hundred  and  1  hun- 
dredth ?  What  is  the  use  of  the  0  in  decimal  notation  ?  of  the 
naught  prefixed  ?  annexed  ? 

5  $  53,  54.  How  do  you  express  decimals  in  figures  ?  Give 
an  example.  How  do  you  read  decimals  expressed  in  figures  ? 
Give  an  example. 

5  55.  Is  the  rule  for  the  addition  of  decimals  in  any  wise  simi- 
lar in  principle  to  the  rule  for  the  addition  of  common  fractions 
and  of  whole  numbers  ?  Why  must  as  many  figures  in  the  prod- 


324  APPENDIX.  [CHAP.  x. 

uct  be  pointed  o.7  as  are  equal  to  the  greatest  number  of  decimal 
places  in  any  of  :  iie  numbers  added  ? 

§56.  Is  the  principle  in  subtraction  of  decimals  like  that  in 
subtraction  of  common  fractions  and  of  whole  numbers  ?  Show 
this.  How  must  the  difference  be  pointed  off? 

\  57.  Give  an  example  of  the  right  mode  of  pointing  off,  in  the 
product  of  one  decimal  by  another,  with  reasons.  Show  the  ser- 
vice of  prefixing  naughts  to  the  product  in  certain  cases. 

§58.  How  may  a  decimal  number  be  multiplied  by  10,  100, 
1000,  &c.  ? 

\  \  59j  60j  61,  62.  Give  the  rule  for  pointing  off  the  quo- 
tient in  division  of  decimals.  Give  the  reason  for  the  rule. 
When  is  the  quotient  a  whole  number  ?  When  there  is  a  re- 
mainder, what  is  to  be  done  ?  What  does  the  sign  -f-  mean  an- 
nexed to  a  quotient  ?  How  do  you  divide  a  decimal  by  10,  100, 
&c.  ?  Give  rule. 

\  63.  What  is  the  difference  between  a  common  fraction  and  a 
decimal  ?  How  do  you  reduce  a  common  fraction  to  a  decimal  ? 
How  do  tenths  become  hundredths  ?  Explain  the  reason  for  the 
rule.  Are  a  repetend  and  a  repeating  decimal  the  same  thing  ? 
What  is  the  difference  between  them  ?  What  is  the  finite  part  ? 

\  64.  Give  the  reasons  for  the  rule. 

\  65.  How  do  you  reduce  repeating  decimals  to  common  frac- 
tions ?  Give  the  reason  for  the  rule.  What  is  to  be  done  if  there 
be  a  finite  part  to  the  decimal  ? 

§  66.  What  is  Federal  Money  ?  What  is  its  symbol  ?  What 
is  the  difference  between  7'682  and  $7*682  ?  Are  the  denomi- 
nations of  Federal  Money  to  be  explained  by  decimals  ?  If  so, 
how?  Is  the  gold  or  the  silver,  of  which  the  United  States 
coins  are  made,  pure  ?  Why  is  it  not  pure  ?  What  is  the  alloy  ? 

§  67.  Illustrate  how  the  annexing  or  the  cutting  off  of  O's 
changes  the  denominations  of  Federal  Money. 


CHAP.  X.J  APPENDIX.  325 

§  69.  Explain  the  method  and  the  principle  of  dividing  deci- 
mals by  10,  100,  or  1000,  &c. 

§  70.  What  are  aliquot  parts  ?  Explain  the  method  of  obtain- 
ing results  by  aliquot  parts.  Explain  the  rule.  What  is  the 
origin  of  our  denominations  of  shillings  and  pence  ?  Why  are 
the  shillings  of  the  different  States  of  unequal  value  ?  In  what 
year  \vas  Federal  Money  adopted  ? 

ANSWERS. 

*53.  (1-9.)  2;  3;  6;  4;  5;  9;  7;  8;  10.  (1O-19.)  0-037 ; 
0-03 ;  0-000048  ;  0-00000095  ;  0-00490  ;  0*0001240  ;  0-10000004  ; 
0-000000096;  0-00009301;  0-027101.  (2O-27.)  0-0804000; 
7-000400;  0*074000081;  0-00896000;  0-04007;  0*0800004000; 
0-060000000074;  80-0000000099.  (28-37.)  0'84 ;  0-096; 
0-0077;  0-00104;  0-10007;  0-000044;  0*0000007;  0-0000012; 
0-01365.  (38-50.)  9-4;  8-13;  41-8;  418-9;  46-74;  8-961; 
0-7461;  54-982;  4786-19;  2-826018;  18-9765;  8-4108; 
8976-54821. 

§54.  (51-78.)  Eight  tenths ;  ninety  hundredths  ;  four  hun- 
dred and  seven  thousandths;  one  thousandth;  six  thousand  nine 
hundred  and  forty-five  ten  thousandths ;  eighty-seven  thousand 
six  hundred  and  one  hundred  thousandths ;  seventy-six  hundred 
thousandths ;  ten  thousand,  eight  hundred  and  seventy-six  hun- 
dred thousandths ;  one  thousand  and  seven  ten  millionths ;  one 
million  and  twelve  ten  millionths ;  six  million,  seven  hundred  fifty 
thousand,  nine  hundred  and  twelve  ten  millionths;  eighty  million, 
seven  hundred  thousand,  one  hundred  and  seventy-six  hundred 
millionths  ;  eighty  million  and  one  hundred  millionths;  nine  hun- 
dred and  one  million,  ten  thousand,  one  hundred  and  one  bil- 
lionths ;  three,  and  seventeen  ten  thousandths ;  four,  and  ninety 
thousand  and  eighteen  hundred  thousandths ;  six,  and  one  mil- 
lionths; forty-nine,  and  one  hundred  thousand  and  seven  mil- 
lionths ;  eighty-six,  and  ten  thousand  and  seven  ten  millionths ; 
forty-four,  and  sixty-two  million  and  sixteen  hundred  millionths  ; 
one  billion,  one  million  and  one  hundred  ten  billionths ;  twenty- 
seven,  and  forty-six  thousand,  eight  hundred  and  twelve  hundred 

28 


326  APPENDIX.  [CHAP.  x-. 

thousandths;  nine,  and  four  hundred  and  sixty-seven  ten  thou- 
sandths ;  eight  and  forty-two  ten  thousandths ;  twenty-one,  and 
one  ten  millionths ;  thirty-six,  and  twenty-one  and  one-sixth  hun- 
dredths ;  forty-eight,  and  four  thousand  eighty-one  and  one-ninth 
ten  thousandths ;  nine,  and  ten  million,  one  hundred  one  and  sixty- 
nine  seventieths  of  hundred  millionths. 

§55.  (79-81.)  0-8145;  1-1718;  561843-617.  (82-§7.) 
1-4031;  1-1655;  1-3464;  1-9998;  2-5587;  3-1718.  (8§-93.) 
17-2725;  0'42111;  532-94;  170-972;  367-4221;  2l57'114. 
(94-98.)  642-206;  2-1769;  1342-048;  2523-315;  410-714. 
(99-1O3.)  72527-835;  260*557;  317-809;  6507*42057; 
127944-077. 

§56.  (1O4-11O.)  850-7973;  197-7603;  861*4708  ;  898*- 
2632;  897-7603;  1-0809;  896'2994.  (111-116.)  84120-21  ; 
92487-3091  ;  92580-614184  ;  10580-31;  92581-309964;  92539'- 
812.  (117.)  2999999-900001.  (118.)  96000002006*9999916. 
(119.)  82000302-9999L  (12O-122.)  291-222;  905'3356 ; 
2977-02313346.  (123-125.)  923-5;  0'34803 ;  0'10557. 
(126-129.)  208-3263  ;  3652*005  ;  203-3735  ;  495-7564. 

§57.  (13O-138.)  7544*624;  11316-936;  13203-092; 
26406-184;  36780-042;  114112-438  ;  656382-288  ;  1169416-720; 
63065174-696.  (189-147.)  567*96621;  1917-562109;  1623*- 
5719803;  25099-7791204;  19-7435873;  79'67754546 ;  256*- 
450266820;  207-1510702521;  276867-86773521.  (148-154.) 
17-07577096;  245-0806812;  18-43085209;  3-09751462586; 
0-00000298806;  0-000000448209;  0*00000000398408.  (155- 
163.)  0-0001406;  0*00002072;  0-000002664;  0'0000003552  ; 
0-00000007104;  0*000000006216  ;  0-0000000007178  ;  2*699594; 
6-612640296. 

§58.  (164.)  821*46.  (165.)  76920.  (166-171.) 
46104  ;  461040  ;  4610400  ;  46104000  ;  461040000;  4610400000. 
(172-179.)  4-7692  ;  47*692  ;  476*92  ;  4769*2 ;  47692  ; 
476920;  476900;  47692000.  (18O-187.)  37;  370;  3700; 
37000:  370000;  3700000:  37000000;  370000000. 


CHAP.  X.]  APPENDIX.  327 

559.  (188-192.)  2-3;  3;  11-2;  2'6;  56-71457.  (193- 
198.)  3-4024;  0*456;  1*1213  ;  9-8008;  9-6665;  38-666. 

I  6O.  (199-205.)  10;  500;  5000;  25000;  36;  198;  40. 
(206-213.)  471-73+  ;  832-07+  ;  117-11+ ;  29-68+  ; 
10-24+;  5-58+;  1-05+;  68-46.  (214-225.)  100931*7465+; 
91654-520+  ;  725003-53+  ;  2530*6+;  114+;  84+  ;  13090+ ; 
8954+  ;  20126+  ;  37-940+  ;  98528-6097+;  85314-94+. 

§  61.  (226-231.)  1-643  ;  8-235  ;  0'1647;  0-1176+;  0-0915  ; 

0-0748+.     (232-238.)    6666-666+;    8-3111+ ;    2'2886+  ; 
0-824+  ;  2-223+  ;  7-405+  ;  22-629. 

§62.  (239-242.)  4149-76;  414-976;  41*4976;  4-14976. 
(243-247.)  6-74  ;  0'674  ;  0-0674  ;  0'00674  ;  0-000674.  (248- 
253.)  0-0341  ;  0-00341  ;  0'000341  ;  0'0000341  ;  0*00000341  ; 
0-000000341.  (254.)  3'95  cords.  (255.)  13*2  inches, 
(256.)  88-28  inches.  (257.)  63-38  nearly.  (258.)  103-94+ 
volumes.  (259.)  320  rods.  (26O.)  498-65+  seconds. 
(261.)  70000-039492  grains  =  10-000005  pounds,  nearly. 
(262.)  2218-192  cubic  inches  ==  1-283  cubic  feet,  nearly. 
(263.)  436247-424  grains=62-32106  pounds,  nearly.  (264.) 
37059-62+  times.  (265.)  320  rods.  (266.)  52'8  feet. 
(267-269.)  3462-29  times,  nearly;  2974-65  times,  nearly; 
487-64  times,  nearly.  (27O.)  54-01+  dollars.  (271-274.) 
2400-375  dollars  ;  80  dollars  ;  99-625  dollars  ;  12  acres.  (275.) 
584-75  gallons. 

§63.  (276-297.)  0-5;  0-333+ ;  0-14285714+ ;  0-1666+ ; 
0-08333+  ;  0-666+  ;  0'75  ;  0-8  ;  0'8333+  ;  0-85714285+  ; 
0-875  ;  0-888+  ;  0*8181+  ;  0-90909+  ;  0'91666+  ;  0-92857+  ; 
0-9333+ ;  0-9375;  0-941 17+;  0-9444+ ;  0-94736+ ;  0'95. 
(298-329.)  0-6 ;  0-42857+  ;  0-375  ;  0'3  ;  0-230769+  ; 
0-571428+ ;  0-4444+ ;  0-3636+ ;  0-307692+ ;  0-71428+ ; 
0-625;  0-5555+;  0-4545+ ;  0-5454+ ;  0-461538+;  0-352941+ ; 
0-7777+;  0-6363+;  0-53846+;  0-7272+  ;  0-61538+;  0'5333+; 
0-470588+  ;  0-692308+ ;  0-52941+  ;  0*785714+  ;  0-590909+  ; 
0-57142£+  ;  0*575757+;  0-921568+  ;  0*734513+  ;  0*160112+ 


328  APPENDIX.  [CHAP.  x. 

(330-333.)  0-685714+  ;  0-31818;  0*85069+  ;  0*349205+  . 
(334-336.)  3-256277+;  0'080148+;  1  '256493+.  (337.) 
15-716666+  dollars. 

$64.  (338-351.)    f5o=l;  !*&=&;  *=i;  iVA-i; 

225    _   9     .         4:15  _   87     .         575   _  23.  486  -  243  .          656   -   82     . 

TinnF  -  40'      TOTO  -  2~0~0~  »       TO"00  -  4  0"  J        TTToTr  —  31Tff  '        ToW  -  125~> 

25          _1       ,          375      __3       .  225        _        9          .       1001.36984    __ 

TTlFoD  —  ¥oT  '   ToTooT  —  So'fF  '    TZFTFcToinF  —  ¥7oT7T  '   TooTo"  >  nFffoinr  — 

rWoV     (352-357.)  y£Ji*=;^&T>  TiWW--  iJHrJ  Tiffs  ?nr— 

1    .   5005  _  1001.    125  _  1  .    1250505  —  250101 

>  roo^oo  —  5fro~o  >  TTTCTO?  —  so"'  To"ooo(rooo"  —  ^ 


(358-370.)    J;   TVA  5   TVr  5   rWr  5  *5  A;  A? 

TV;  i  i  A  ;  iW-  (371-385.)  ^VVVV  ;  A  ;  A  ;  A  ; 

55  J  '  TT»  TT'  ¥36°»89"J  T7T°T5r'  TTT  5  TTTT  >  jViT  >  TTJ  IT' 


^  66.  (386-392.)  Seven  dollars  and  eighty-four  cents  ;  nine- 
ty-two dollars  and  six  cents  ;  six  hundred  and  seventy-two  dollars 
twelve  cents  and  three  mills  ;  eight  thousand  nine  hundred  and 
sixty-one  dollars  and  six  mills  ;  four  thousand  one  hundred  and 
eighty  dollars  ninety-six  cents  and  seven  and  three-tenths  mills  ; 
nine  hundred  and  one  dollars  and  one  mill  ;  three  dollars  and  three 
cents.  (393-4O1.)  Six  dollars  and  eighty-two  cents;  seven 
dollars  forty-four  cents  and  eight  mills;  nine  dollars  and  two 
cents  ;  three  dollars  and  one  cent  ;  four  dollars  and  seven  cents  ; 
six  dollars  and  ninety-three  cents  ;  forty-eight  dollars  seventy-six 
cents  and  one  mill  ;  two  hundred  and  seventeen  dollars  and  one 
mill;  thirty-six  dollars  ninety-eight  cents  and  seven  mills. 
(4O2-4O§.)  $0-37;  $0'443;  $6'02;  $4-008;  $9-206; 
$5000-089;  $1000000-015.  (4O9-412.)  $0'375  ;  $2-125; 
$4-625;  $5-875. 

§67.  (413-419.)  800c*.=8000  mills  ;  89400cZ.=:894000m.; 
62000^.  =  620000m.  ;       3400c£.  =  34000m.  ;       93627300c£.  = 
936273000m.  ;      84190400cZ.  =  841904000m.  ;      12345600cZ.  = 
123456000m.     (42O-426.)    830m.  ;    910m.  ;    40m.  ;  3780m. 
12340m.  ;  91000m.  ;  8756180m.    (427-441.)    $8'41  ;  $9-28 
$46-70  ;  $129-86  ;  $4-81  ;  $1-234  ;  $49'68  ;  $321*946  ;  $1357'92 
$9-80;    $98;    3918-762;    £49876-21;    $30760-09;    $4876-543. 


CHAP.  X.]  APPENDIX.  329 


§  68.  (457.)  -$20-80.  (458.)  $99'05.  (459.)  $82'288. 
(460.)  $106-195.  (461.)  $6'375.  (462.)  $25'53.  (463.) 
$20-9375.  (464.)  $49-815.  (465.)  19625'796-f.  (466.) 
$19597-125.  (467.)  $1'929+.  (468.)  $68-493+.  (469.) 
216-5454+  rods.  (47O.)  $38-861+.  (471.)  $11428-571. 
(472.)  7682  pounds.  (473.)  7897  pounds  sold  ;  average  for 
each  cow,  149  pounds.  (474.)  21631-5  pounds,  $3244'725. 
(475.)  $1497-9225.  (476.)  $12-149+.  (477.)  1575  thou- 
sand m's ;  he  received  $236-25.  (478.)  $16'81.  (479.) -Butcher 
receives  from  tailor  $19-77,  receives  from  shoemaker  $15-14,  and 
the  tailor  receives  from  shoemaker  $20-24.  (48O.)  $4*078125. 
(481.)  $41-535.  (482.)  $128'52.  (483.)  $27'0225.  (484.) 
$122-6475.  (4§5.)  $0'625.  (486.)  $2164781.  (487.)  $133'36. 
(488.)  $94-27.  (489.)  $16'665.  (49O.)  1314  volumes. 
(491.)  $2299-50.  (492.)  $335'73. 

I  69.  (493.)  $7-15.  (494.)  $16-875.  (495.)  $248-2875, 
(496.)  $339-0825.  (497.)  $2-84375.  (498.)  $1351-3728. 
(499.)  $8538-75.  (5OO.)  $2090-0376. 

57O.  (5O1-514.)  $84-75;  $113;  $127-125;  $135-60; 
$169-50;  $211-875;  $226;  $254'25  ;  $339;  $423-75;  $452; 
$508-50;  $550-875;  $593-25.  (515-521.)  $105  ;  $315  ;  $420  ; 
$525;  $630;  $735;  $210.  (522-527.)  54  pounds;  72 
pounds ;  324  pounds ;  5556  pounds  ;  49302  pounds ;  588762 
pounds.  (528-538.)  592  pecks  ;  296  pecks  ;  222  pecks  ;  444 
pecks;  49i  pecks;  88£  pecks;  63f  pecks;  40T\  pecks;  74 
pecks;  55|  pecks;  44f.  (539-549.)  800  brushes;  400; 
266f ;  200;  160;  133^:  114^;  100;  80;  66f ;  57  j.  (55O- 
562.)  17491  yards;  1312;1166f;  1124f;  1049§  ;  984  ;  874f  ; 
787};  583^;  524f ;  492;  437^;  403  &.  (563-575.)  $85'75  ; 
$102-90;  $104-125;  $106'66|;  $110-25;  $114-33^;  $122-50; 
6130-66|;  $159-25;  8134-75  ;  $153-125;  $147;  $183-75. 

28* 


330  APPENDIX.  [CHAP.  XL 


CHAPTER  XL 

§  71.  Explain  the  difference  between  an  abstract  and  a  denom- 
inate number.  Which  of  the  two  is  Federal  Money  to  be  con- 
sidered ?  Which,  any  multiplier  ?  Which,  the  product  of  two 
numbers  ?  When  is  a  quotient  to  be  considered  an  abstract  num- 
ber, and  when  a  denominate  number  ?  Give  an  example. 

§  72.  What  is  meant  by  Sterling  Money  ?  Is  the  pound  in 
circulation  ?  What  do  the  symbols  £,  s.,/.,  d.,  qr.,  express  ? 

§  73.  What  was  the  original  of  all  weights  ?  Whence  the 
term  Troy  ?  What  are  weighed  by  this  weight  ? 

§  74*  Draw  upon  the  board  the  symbols  of  the  grain,  the  dram, 
the  scruple,  the  ounce,  the  pound,  Apothecaries'  Weight  ?  For 
what  is  this  weight  used  ? 

§  75.  Wherein  does  Avoirdupois  Weight  differ  from  Troy  ? 
What  is  said  concerning  the  qr.  ? 

§  76.  Whence  was  our  standard  yard  obtained  ?  What  is  the 
French  standard  or  unit  of  measure  ?  How  was  it  obtained  ? 
How  is  the  inch  often  divided  ?  How  on  the  carpenters'  rules, 
which  you  commonly  see  ?  What  is  the  difference  between  a 
knot  and  a  nautical  mile  1 

\  77.  Repeat  the  table  of  Cloth  Measure. 

\  78.  Explain  what  is  meant  by  Square  Measure.  Is  an  acre 
square  ?  Why  not  ?  How  long  is  Gunter's  chain  ? 

§  79.  What  is  the  difference  between  square  and  solid  meas- 
ure ?  between  a  square  foot  and  a  solid  foot  ?  Give  examples. 
What  is  meant  by  round  timber  ?  What  is  a  cord  foot? 

§  8O.  Repeat  the  table  of  Wine  Measure. 

§  81.  Which  is  the  larger,  the  wine  or  the  beer  gallon  ?  the 
wine  or  the  beer  quart  ?  How  much  larger  ?  In  which  should 
milk  be  measured  ? 


CHAP.  XI.J  APPENDIX.  331 

o  82.  Repeat  the  table  of  Dry  Measure.  What  articles  not 
mentioned  in  the  note  are  measured  by  this  measure  ?  What  is 
the  U.  S.  standard  of  Dry  Measure  ? 

§  83.  What  is  leap  year  ?  What  is  the  centennial  year  ? 
What  is  the  difference  between  a  lunar,  a  calendar,  and  a  business 
month  ? 

5  84.  Divide  a. circle  in  halves  by  a  straight  line.  How  many 
degrees  in  each  half?  How  many  in  a  quarter-circle  ?  Explain 
latitude — longitude.  How  many  miles  in  a  degree  ?  How  many 
miles  does  the  sun  pass  over  in  an  hour  ? 

§  85.  Repeat  the  table. 

§  86.  Explain  the  difference  between  Reduction  Ascending 
and  Reduction  Descending.  Repeat  the  rule,  and  apply  it  to  each 
of  the  four  cases  of  reduction  descending,viz. :  (1st.)  of  a  compound 
quantity  toils  lowest  denomination;  (2d.)  of  the  fraction  of  a 
higher  to  the  fraction  of  a  lower  denomination ;  (3d.)  of  the  frac- 
tion of  a  higher  to  its  value  in  lower  denominations ;  (4th.)  of  the 
decimal  of  a  higher  to  its  value  in  lower  denominations. 

§87.  Give  the  rule  for  Reduction  Ascending.  Specify  the 
four  cases  to  which  it  may  be  applied,  and  apply  and  illustrate  it. 

5  88.  Wherein  does  the  principle  involved  in  addition  of  de- 
nominate numbers  differ  from  that  of  simple  addition,  and  of  ad- 
dition of  fractions  ?  Give  rule  and  explain  it. 

§  89.  Why  must  a  number  of  the  subtrahend  be  placed  under 
a  number  of  the  same  denomination  in  the  minuend  ?  What  dis- 
tinct principles  are  embodied  in  the  rule  ? 

§  SO.  Give  the  rule  and  explain  its  principles. 

5  91.  Show  in  what  way  a  divisor  may  be  an  abstract  or  a  de- 
nominate number.  Show  how  a  quotient  may  be  an  abstract  or  a 
denominate  number. 


332  APPENDIX.  [CHAP.  XL 

§  92.  What  is  the  meaning  of  Duodecimals  ?  What  are  their 
denominations  ?  To  what  are  they  generally  applied  ?  What  is 
there  peculiar  in  the  addition  or  subtraction  of  duodecimals  ? 

§  93.  Illustrate  the  multiplication  of  duodecimals  by  the  multi- 
plication of  decimals.  What  may  the  index  (')  be  considered  ? 
What  is  the  rule  for  the  annexing  of  indices  to  the  product  ? 
Explain  this.  What  is  the  strict  value  of  1'  in  the  measurement 
of  surfaces  ?  What  in  the  measurement  of  solids  ?  What  as  a 
linear  measure,  for  which  it  is  sometimes  used  ? 

§  94.  Give  an  instance  of  the  practical  use  of  division  of  duo- 
decimals. Go  through  with  an  operation,  applying  the  rule  and 
explaining  each  step.  How  can  you  illustrate  the  indices  of  the 
quotient  by  the  decimal  places  in  decimal  division  ? 

§  §  95.  96.  In  the  addition  or  suotraction  of  denominate  frac- 
tions, what  principle  is  involved  different  from  that  in  addition 
and  subtraction  of  common  fractions  ? 

ANSWERS. 

$85.  (1-10.)  8;  16;  20;  32;  40;  60;  80 ;  100;  200; 
400.  (11-20.)  48;  96;  144;  240;  384;  720;  960;  1200; 
2400;  4800.  (21-3O.)  24;  36;  60;  84;  108;  180;  240; 
300;  600;  1200.  (31-4O.)  40;  60;  100;  140;  180;  300; 
420;  500;  1000;  2000.  (41-42.)  6£ ;  1050£.  (43.)  1 
8overeign==20s.=240d.=960/0r.  (44.)  498.  (45-GO.)  48; 
120;  168;  360;  600;  1200;  480;  960;  2400;  3360;  7200; 
12000;  24000;  5760;  11520;  28800.  (61-74.)  40;  100; 
140;  180;  300;  400;  500;  1000;  240;  720;  1200;  2160; 
3600;  6000.  (75-§l.)  24;  48;  108;  180;  300;  600;  1200. 
(82-1O1.)  60;  100;  140;  240;  360;  60;  300;  420;  720; 
1080;  480;  2400;  3360;  5760;  8640:  5760;  28800;  40320; 
69120;  103680.  (1O2-I1O.)  6;  15;  27;  24;  120;  216;  288; 
1440;  2592.  (111-118.)  16;  40;  56;  96;  576;  864;  1440; 
1536.  (119-128.)  32;  1125  144;  2^6;  512;  2304;  5120 ; 
6400;  25600;  512000.  (129-130.)  32;  112;  144;  240;  400; 


CHAP.  XI.]  APPENDIX.  333 

1600  ;  1600  ;  32000.  (137-141.)  2000  ;  6000  ;  40000  ;  100000  ; 
200000.  (142-15O.)  24;  84;  240;  36;  198;  198;  7920; 
63360;  4382400.  (151-153.)  300  ;  5280  ;  528000.  (154- 
15§.)  72;  63;  360;  72;  1440.  (159-164.)  36;  9;  27; 
37£;  45;  54.  (165-167.)  6048;  1568160;  6272640. 
(168-169.)  4840;  3097600.  (170-172.)  46656;  86400; 
221184.  (173-177.)  24 ;  48 ;  1512;  6048;  12096.  (178- 
180.)  504;  252;  2016.  (181-182.)  36;  72.  (183-186.) 
32;  1024;  15360;  25600.  (187-189.)  200;  6400;  12800. 
(19O-197.)  300;  900;  1800;  2700;  3600;  43200;  86400; 
604800.  (198-2OO.)  168;  8736;  8760.  (2O1-2O7.)  120; 
480;  900;  7200;  28800;  54000;  1296000.  (2O8-211.)  120; 
720;  1500;  21600.  (212-214.)  72;  864;  10368.  (215- 
218.)  100  ;  500 ;  2000  ;  4000.  (219-221.)  980  ;  4900  ;  19600. 
(222-225.)  120;  480;  3360;  3840.  (226-233.)  25;  24; 
12;  28;  90;  6;  9;  18.  (234-239.)  12;  40;  8;  12;  10;  8. 
(240-246.)  18;  12;  3;  20;  32;  31;  5.  (247-25O.)  5; 
37|;  18f  ;  10.  (251-262.)  4;  121 ;  4£ ;  41f ;  72  ;  5  ;  50; 
101;  20;  30T% ;  60;  14£.  (263-265.)  7;  15;  33|.  (266- 
268.)  6;  41;"  75.  (269-272.)  12;  30T% ;  81;  381.  (273- 
276.)  20;  50;  41;  39.  (277-281.)  100;  22J;  31' ;  22f|; 
44TV  (282-286.)  44;  200;  309;  81;  621.  "(287-289.) 
150;  120;  100.  (29O-292.)  640;  2560;  102400.  (293- 
295.)  4;  330;  10.  (296.)  10.  (297.)  12-J.  (298.)  25. 
(299-300.)  5;  16|. 

§  87.  (301.)  45369.  (3O2.)  £123  10s.  Id.  3far.  (303.) 
1500.  (304.)  234.  (305.)  1214.  (306.)  Sib.  5oz.  ISpwt, 
Ugr.  (307.)  15/6.  Ooz.  5pwt.  (3O8.)  83/6.  4oz.  (3O9.) 
26237.  (310.)  576001.  (311.)  425.  (312.)  2^  03  23. 
(313.)  2Ri  03  03  03  Igr.  (314.)  IT.  Ucwt.  Oqr.  11/6. 
lOdr.  (315.)  8340.  (316.)  512257.  (317.)  IT. 
Iqr.  21/6.  7oz.  14f/r.  (31§.)  120yd.  2ft.  11  in.  (319.)  15713280. 
(32O.)  1577664000.  (321.)  288.  (322.)  58.  (323.)  133. 
(324.)  16.  (325.)  62yd.  2qr.  (326.)  243.  (327.)  1440. 
(32§.)  307200P.  (329.)  138030J.  (33O.)  2R.  15P.  161| 
sq.ft.  (331.)  7.  (3SS2.)  500000.  (333.)  2176.  (334.) 


APPENDIX.  [CHAP,  xi 

1175040.  (335.)  207  tons,  8  cubic  feet,  1721  cubic  inches 
(336.)  125.  (337.)  32832.  (33§.)  32256.  (339.)  9  bar 
rels,  29  gallons.  (34O.)  3786.  (341.)  31  barrels,  15  gallons, 
3  quarts.  (342.)  25.  (343.)  4752.  (344.)  231/iArf.  ZQgal. 
(345.)  4323.  (346.)  2304.  (347.)  26528.  (348.)  4ch. 
24bu.  Ipk.  (349.)  49/;w.  3pk.  6qt.  (35O.)  2592000.  (351.) 
166554.  (352.)  789458400.  (353.)  6731 55da.  IShr.  (354.) 
9496da.  (355.)  11  birthdays.  (356.)  388800''.  (357.)  189°.' 
(358.)  16°  40'.  (359.)  2°  46'  40".  (36O.)  164735". 
(361.)  87i</oz.  (362.)  76  buttons.  (363.)  555  gross,  6§doz. 
(364.)  70  years.  (365.)  360  sheets.  (366.)  480  sheets. 
(367.)  IT'™-  (368.)^.  (369.)^-  (37O.)  1760  feet. 
(371.)  w^ff.  (372.)  $.  (373.)  5971  gills.  (374.)  34f 
inches.  (375.)  TJ,YF.  (376.)  ffJT.  (377.)  129f|  inches^ 
10££  feet=3j$  yards.  (378.)  ^fa  of  a  month^^ir  of  a  year. 
(379.)  Iqr.  2?-n«.  (380.)  1/ur.  20rrf.  (381.)  Igr.  42|ZA. 
(382.)  2mi.  3>r.  26rrf.  ll//!.  (383.)  9hr.  36min.  (384.)  5min. 
37^sec.  (385.)  5/tr.  48min.  48sec.  (386.)  13oz.  2f^|^r. 
(387.)  -3J7/.  (388.)  30rfa.  (389.)  25P.  (39O.)  £8-259375. 
(391.)  0-875  of  a  yard.  (392.)  0-4444+  of  a  yard.  (393.) 
3-36684027777+  pounds.  (394.)  10-3995265151+  miles. 
(395.)  0-145949074074+  of  a  day.  (396.)  £3-2520833+. 
(397.)  0-4444+ of  a  hogshead.  (398.)  0-2270833+.  (399.) 
£0-915625.  (400.)  0-921875  of  a  bushel.  (4O1.)  0-469618055+ 
of  a  day.  (4O2.)  0-5219696+  of  a  furlong.  (4O3.)  0'71  of 
an  hour.  (4O4.)  0*0827617+  of  a  year.  (4O5.)  G'24224  of  a 
day.  (406.)  3/2.  25'2P.  (4O7.)  2s.  6d.  (4O8.)  13*.  4d. 
(4O9.)  23g-aZ.  2qL  \pt.  (41O.)  44cfo.  5hr.  49min.  l'632sec. 
(411.)  5oz.  5'888^r.  (412.)  6cwt.  2qr.  Ulb.  4'8oz.  (413.) 
2/wr.  On/.  4yd.  \ft.  2'4m.  (414.)  6s.  10^.  3'2/ar.  (415.) 
2ftr.  54mm.  32'7168sec.  (416.)  6s.  10^.  3'776/ar.  (417.) 
19s.  9d.  (418.)  5hr.4.Smin.  49'536sec.  (419.)  648.  (42O.) 
15.  (421.)  54-32  pounds.  (422.)  67-3064  pounds.  (423.) 
151Z6.  8or.  (434.)  23-64  sheets.  (425.)  11  quires  2-4  sheets. 

§88.    (426.)  £39  15s.  Q$d.     (427.)  £25  4s.  5£d.     (428.) 
£34  14s.  8d.    (429.)   17ZA.  3oz.  Wpwt.  5gr.    (43O.)  21Z6.  lOoz. 


CHAP.  XI.]  APPENDIX.  335 


3pwt.  20gr.  (431.)  61/6.  lloz.  IZpwt.  8gr.  (432.)  321b  85 
03  23  120-r.  (433.)  361b  65  53  13.  (434.)  183  09  13.gr. 
(435.)  27  T.  9cM>Z.  2?r.  21Z6.  7oz.  13dr.  (436.)  llcwt.  Oqr. 
.  7oz.  (437.)  14L.  Omt.  5>r.  15rd.  2yrf.  (438.)  22rd. 
0/2.  8m.  (439.)  58yd.  Iqr.  (44O.)  47£.  Fl.  Oqr.  2na. 
(441.)  52£.  E.  2qr.  2na.  (442.)  17657.  yd.  7  sq.ft.  93sq.  in. 
(443.)  23M.  106A.  012.  34P.  (444.)  26s.  yd.  \8s.ft.Q63s.  in. 
(445.)  35  cords  4.1s.  ft.  (446.)  50  cords  5c.  /*.  (447.) 
54hhd.  36gal  Iqt.  \pt.  (448.)  36^ns  Qpi.  Ihhd.  I9gal  Oqt.  Ipt. 
Igi.  (449.)  56hhd.  Wgal.  Iqt.  Ipt.  (45O.)  33bar.  Sgal.  3qt. 
(451.)  22cL  236w.  2^/c.  4qt.  Ipt.  (452.)  42iw.  l^L  Iqt. 
(453.)  &2da.  21hr.  2min.  9sec.  (454.)  8wk.  6da.  6Jir.  50min. 
33sec.  (455.)  4cr.  Os.  11°  59'  26".  (456.)  9s.  8°  45'. 
(457.)  28°  55'  58". 

§  89.  (458.)  3T.  newt.  Iqr.  24Z6.  4oz.  13dr.  (459.)  59A. 
2R.  27P.  (46O.)  7ib  85  63  13  19^-r.  (461.)  llL.  2mi.  0/ur. 
31rrf.  (462.)  5E.  Fr.  3qr.  Ina.  (463.)  20cA.  Ibu.  Ipk.  2qt. 
Ipt.  (464.)  9/ims  Ipi.  Ohhd.  52gal  2qt.  (465.)  40^a.  9Ar. 
19mm.  16sec.  (466.)  13yr.  7?no.  Owk.  2da.  (467.)  19mi'. 
0/wr.  Ird.  (468.)  34C.  127s.  /i.  (469.)  19C.  1  cord  ft. 
(47O.)  £19  9s.  3d.  (471.)  llmo.  5da.  (472.)  4mo.  28rf«. 
(473.)  2yr.  llmo.  Hc/a.  (474.)  Syr.  IQmo.  4da.  (475.) 
64yr.  Imo.  26tZ«.  (476.)  74yr.  4mo.  28Ja.  (477.)  £83  4s. 
2rf.  (478.)  36c^.  3yr.  19Z6.  602.  (479.)  320A.  I.R.  15P. 
(48O.)  5yr.  3mo.  7Ja.  ;  1925^a.  (481.)  5cwt.  Sqr.  9lb. 
(482.)  %0yd.  2qr.  3na.  (483.)  70  C.  97s.  ft.  (484.)  976w. 
2pk.2qt.  Ipt.  (485.)  11186Z6.=5'593T.  (486.)  45r^.  6%  ft. 
(487.)  124yd  S^r.  Inc.  (488.)  233  cubic  feet=Uc.ft.,  9  cu- 
bic ft.—  1C.  6c.  /i.  Qcubicft.  (489-491.)  3yr.7mo.27Ja.; 
H8yr.  lOmo.  9^a.  ;  llo^r.  2mo.  12da.  (492.)  286yr.  8mo.  8da. 
(493.)  123yr.  2mo.  IBda.  (494.)  13°  38'  30".  (495-496.) 
10°  45'  10";  2°  53'  20".  (497-499.)  1°  38'  43";  12°  23' 
53";  1°  14'  37".  (5OO.)  March  8th,  1502.  (5O1.)  128yr. 
2mo.  6da. 

§9O.  (502-507.)  £31   12s.  6d.  ;  £52  14s.  2cf.  ;  £63  5x.  ; 


336  APPENDIX.  [CHAP.  xi. 

£13  15s.  lOd.  ;  £84  6s.  Sd.  ;  £94  17s.  6d.  (5O§-514.)  24cwt. 
O^r.  Gib.  I2oz.  I5dr.  ;  32cwL  Oqr.  9/6.  loz.  4dr.  ;  40cwt.  Oqr.  lllb. 
5oz.  9dr.  ;  48civt.  Oqr.  13/6.  9oz.  14dr.  ;  56civt.  Oqr.  lllb.  I4oz. 
3dr.  ;  Q4cwL  Oqr.  20/6.  2oz.  Sdr.  ;  12cwt.  Oqr.  22/6.  6oz.  I3dr. 
(515-523.)  24cw/.  Oqr.  18/6.  loz.  5dr.;  32cwt.  Oqr.  24/6.  loz. 
I2dr.-,  40cwt.  Iqr.  5/6.  2oz.  Sdr.  ;  48cwt.  Iqr.  lllb.  2oz.  lOdr.  ; 
56cwZ.  Iqr.  17/6.  3oz.  Idr.  ;  64cwt.  Iqr.  23/6.  3oz. 
2qr.  4/6.  3oz.  15rfr.  ;  88c^/.  2^r.  16/6.  4oz.  IZdr.  ; 
22/6.  5oz.  4rfr.  (524-529.)  296^a/.  Iqt.  Ipt.  Igi.  ;  356  gal 
2qt.  Ipt.  2gi.  ;  254gal.  4qt.  Opt.  2gi.  ;  458g-a/.  1^.  Opt.  2gi.  ; 
GSlgal.  Iqt.  Ipt.  Zgi.;  4l5gal.  5qt.  Ipt.Zgi.  (53O.)  32c^.  l^r. 
15/6.  (531.)  54yd.  2qr.  3na.  (532.)  14C.  119s.  ^.  (533.) 
4da.  5hr.  4mm.  ZQsec.  (534.)  Imi.  255ft.  10m.  (535.) 
673141^.  lOhr.  44mm.  28jsec.  (536.)  £27  2s.  6d.  (53T.) 
10812a-a/.  Iqt.  Ipt.  (538.)  23  C.  be.  ft.  (539.)  3361//.  Ijtw. 
(54O.)  £17  13s.  3i/.  (541.)  £35  9s.  Wd. 


91.  (542-54§.)  8yrf.  2^r.  Ifna.;  6^.  Iqr. 

a.  ;  4z/c/.  l^r.  ^na.  ;  3^.  2gr.  3na.  ;  3yd.  Oqr.  3%na.  ;  2yd.  3qr. 
(549-562.)  13c?^.  3qr.  22/6.  14oz.  12^r.  ;  9cwt.  Iqr. 
6/6.  I5oz.  3rfr.  ;  6cwt.  3qr.  23/6.  15oz.  6±dr.  ;  5c^.  2^r.  9/6.  2oz. 
^r.  15/6.  I5oz.  9±dr.  ;  3c^.  3qr.  24lb.6oz.  S\dr.  ; 
24/6.  loz.  ll\dr.  ;  3cwt.  Oqr.  10/6.  lOoz.  6^c?r.  ; 
Icwt.  Oqr.  3/6.  8oz.  12jrfr.;  Iqr.  19/6.  8oz.  5^r.  ;  l^r.  9/6.  Soz. 
4^dr.  ;  l^r.  18/6.  lOoz.  15||c/r.  ;  2^r.  16/6.  9ozf  lT\rfr.  ;  2^r.7/6.  Ooz. 
14^r.  (563-56T.)  9oz.  18pi^/.  3}fgr.  ;  6oz.  15^w>/.  14TV^r.  ; 
Soz.  12pivt.  £gr.  ;  4oz.  8pwt.  19f|^r.  ;  4oz.  Zpwt.  2^gr.  (56§.) 
4mi.  2fur.  39rd.  3yd.  Oft.  l^in.  (569-5T2.)  39gal.  6pt.; 
IA.  OR.  IP.',  Igi.;  £3  13s/  4d.  (573.)  365da.  5hr.  48mm. 
48sec.  (574.)  15//.  2m.  (575.)  7s.  6d.  (576.)  15s.  Sd. 
(577.)  2yd  Iqr.  2na.  (578.)  16rd.  before,  and  1/wr. 
after.  (579.)  9/6.  loz.  14j9Mrt.  5gr.  (58O.)  26w.  3pA: 
(5§1.)  72  reams,  6  quires,  and  2  sheets.  (582.)  The  widow 
had  £856  13s.  4d.,  and  each  child  had  j£244  15s.  2d.  3%  far. 
(583.)  16A.  3R.  8P.,  21A.,  21A.,  42A.  (584.)  6cwt.  2qr. 
16/6.,  2cwt.  Oqr.  24/6.,  Icwt.  Oqr.  12/6.,  3cwt.  Iqr.  8/6.  (585.) 
Soz.  Spirt.  8fgr.  (586.)  1/6.  3oz.  Spiot.  l5T\gr.  (587.)  3424;95 


CHAP.  XI.]  APPENDIX.  337 

gr.  of  gold,  190-275gr.  of  silver,  190-275gT.  of  copper.  (588.) 
$248-062.  (5§9..)  $15'5l5.  (590.)  27?/r.  Qmo.  3wk.  2da.  23/ir. 
42mm.  30\sec.  (591.)  4yr.  Imo.  llda.  in  Southern  States; 
3yr.  Qmo.  14da.  in  Western  States  ;  2yr.  7mo.  24cfo.  in  Northern 
Slates;  8mo.  20da.  in  Middle  States:  llyr.  Qmo.  15da.  in  all. 
(592.)  365242da.5hr.13min.8sec.  (593.)  6939'55<fa.  (594.) 
13°  10'  34"-8+.  (595.)  £43  10s.  Qd.  Iqr.  (596.)  3506/6. 
9oz.  (597.)  11-857+  miles.  (598.)  989*0833+  miles.  (599.) 
il*s.  (600.)  86687500o-aZ. 


$  92.  (601.)  103/.  1'  3".  (6O2.)  2224/.  0'  3"  5"'. 
(603.)  3  If.  9r  9/;.  (6O4.)  547/.  3'  8".  (6O5.)  5  1/.  6'  7". 
(6O6.)  10/.4'7".  (607.)  6/.  8'  8".  (6O8.)  4  1/.  9'  llr/. 
(6O9-612.)  5  1/.  1'2";  24/.  9'4r/;  131/.  8'7r/;  67/.  10'  11". 
(613-615.)  88/.  0'  II";  76/.  1'  10";  64/.  2'  9".  (616- 
619.)  6  1/.  1'  1";  33/.  lf  1"  2'";  121/.  2'  1"  11"';  S2/.  7'  7" 
0"'.  (62O.)  24/.  1'  10"  6'". 

}  93.  (621.)  27/.  0'  7"  9"'  6"".  (622-623.)  48/8""'; 
28/.  3'  11"  2'".  (624.)  2  1/.  1'  9".  (625.)  394/.  2'  9".  (626.) 
3978/.  1'  6".  (627.)  24yd.  6/,  11'  11".  (62§.)  38§£yds. 
(629.)  266/?.  (630.)  ^34'32-h.  (631.)  448  cubic  feet,  T  4". 
(632.)  4610f  cw^ic/!.  (633.)  1655^.  //.  (634.)  $10-903+. 
(635.)  $19-998+.  (636.)  562/.  4'  11"  8"'  1"";  13337/.  9'  9" 
2'"  I""  4'""  T"".  (637.)  31590  bricks. 

\  94.  (638-640.)  2/.  3'  ;  I/.  1'  6"  ;  4/.  6'.  (641.)  2/.  11'. 
(642.)  I/.  11'.  (643.)  8/.  (644.)  SO/  8'.  (645.)  5/.  5'. 


(646-64-8.)  2d.'2jfar;  6qt.l$pt.;  19hr.  5min.  5]§sec. 
(649-650.)  \ft.  lOiin.;  15s.  10d.  2T%^r.  (651-652.)  3da. 
IGhr.  15mm.;  75d«.  2?ir.  (653.)  2^r.  8Z6.  9oz.  5idr.  (654.) 
5&gt.  (655.)  7fm.  (656.)  4da.  21hr.  8mm.  (657.) 
0^.  fyt.  (658.)  $10-1  5|. 

§  96.  (659-66O.)  8hr.  59mm.  48sec.  ;  I8hr.  57mm. 
(661.)  4yd.3qr.mna.    (662.)  21Z.  4f  |».     (663.) 

29 


338  APPENDIX.  [CHAP.  xii. 

(664.)  Qgal.  Sqt.  fp*.  (665.)  2A.  IR.  10-^P.  (666.)  2GgaL 
l^pt.  (667.)  387-29J-  days  of  10/zr.,  counting  365  days  to  the 
year.  (66§.)  3s.  2d  3|/ar.  (669.)  3da.  Mr.  I5sec.  (67O.) 
2oz.  ISpwrt.  3fgr.  (671.)  lOcte.  (672.)  46365c/s.  (673.) 
&V»  &V  AVi  lei-  CO**-)  8^r.  30mm.  (675.)  3Ja.  9ftr. 
llmw.  40sec.  (676.)  $112.  (677.)  $200,  $120,  $880.  (67§.) 
Each  received  576  balls.  (679.)  lOlb.  lloz.  6%dr.,  lllb.  2oz. 
9\dr.,  11/6.  1102.  Sdr.,  12/6.  7oz.  3|dr.,  53Z6.  1602.  4jrfr.  (68O- 
6§1.)  Each  son  received  28A.  3 JR.  36|P.  There  remained  with 
the  father  86A.  3R.  28}P.  (682.)  5da.  6/ir.  (683.)  6s.lO^. 
(684.)  6lb.  2T430Z.  (685.)  274da.  12fer.  30min.  (686.)  39mi. 
5il/wr.  (687.)  Is.  2^ff7Brf.  (688-691.)  Coat  £7  95. ;  panta- 
loons £2  95.  8rf.  ;  hat"  j£l  4s.  IQd.  At  first  he  had  £14  18s. 
(692.)  $1-50.  (693.)  5988|  feet.  (694.)  44'24719-f  inches. 
(695.)  Vlrd.  44ft.  6m.  (696.)  9±in.  (697.)  2yd.  2na.  \\in. 
(698.)  837|  cubic  inches.  (699.)"  223]-  cubic  inches.  (7OO.) 
3763-235  cubic  inches. 


CHAPTER  XII. 

\  97.  What  is  the  principle  of  percentage  ?  Does  percentage 
apply  to  money  alone  ?  Are  common  fractions  used  in  computa- 
tions of  percentage  ?  Why  not  ?  What  is  the  rale  ?  How  do 
you  write  a  rate  that  is  less  than  1  per  cent.  ?  How,  a  rate  great- 
er than  100  per  cent.  ?  What  rate  does  -f^  express  ?  How  is 
the  fractional  part  of  a  number  to  ba  obtained  ?  Give  the  general 
rule  for  obtaining  the  percentage. 

\  \  98,  99.  What  do  you  understand  by  applications  of  per- 
centage ?  What  is  commission  ?  Give  an  example.  What  is 
brokerage  ?  What  are  stocks  ?  What  is  the  par  value  ?  Give 
an  example  of  stock  below  par ;  above  par.  When  will  you  pre- 
fer to  buy  ?  when  to  sell  ?  What  are  dividends  ?  Who  are 
meant  by  stockholders  ?  What  is  the  difference  between  above 
par  and  at  a  discount  1 


CHAP.  XII.]  APPENDIX.  339 

§  100.  If  you  were  an  assessor,  state  how  you  would  proceed 
in  laying  taxes. 

§101.  What  do«you  understand  by  custom-house  business? 
What  officer  of  the  United  States  government  has  the  control  of 
this  department  of  public  affairs  ?  How  are  the  expenses  of  the 
administration  of  government  paid  ?  What  is  revenue  ?  What 
are  duties  ?  What  is  the  difference  between  specific  and  ad  va- 
lorem duties  ?  Wherein  does  net  weight  differ  from  gross  weight? 
Before  computing  duties,  what  allowances  are  made  ?  What  is 
draft  ?  tare  ?  leakage  ?  breakage  ?  What  is  the  legal  allowance 
for  draft  ?  What  is  an  invoice  ? 

§  102.  State  what  you  know  concerning  insurance.  What  is 
the  premium  ?  What  is  meant  by  indemnity  ?  What  is  the  pol- 
icy ?  Are  ships  only  insured  ? 

§  103.  What  is  meant  by  Profit  and  Loss  ?  How  will  yon 
ascertain  the  amount  of  gain  or  loss  in  a  given  transaction  ? 
How  will  you  ascertain  the  per  cent,  gain  or  loss  ?  How  will  you 
ascertain  the  amount  per  yard,  per  bale,  per  pound,  per  piece,  &c., 
gain  or  loss  ?  How  will  you  determine  the  cost  per  yard,  &c., 
knowing  the  sum  for  which  it  sold,  and  the  gain  or  loss  per  cent.  ? 

§  104.  What  is  Interest  ?  Why  called  simple  ?  Wherein  does 
interest  differ  from  percentage  ?  What  has  law  to  do  with  fixing 
the  rate  per  cent,  in  any  transaction  ?  How  do  you  find  the  in- 
terest, if  principal,  time,  and  rate  are  given  ?  How  if  the  rate  be 
6  per  cent.  ?  How  if  the  rate  be  any  per  cent.  ?  If  your  princi- 
pal consist  of  pounds,  shillings,  pence,  &c.,  what  must  be  done  ? 
Explain  aliquot  parts.  Give  an  example. 

5"  105.  Show  a  more  accurate  method  of  finding  the  interest 
when  the  time  is  given  in  days. 

\  106.  What  are  partial  payments  ?  What  is  the  U.  S.  rule 
for  computing  interest  ? 

28* 


340  APPENDIX.  [CHAP.  xn. 

$107,  J  give  you  the  principal,  the  rate  per  cent., "and  the 
time  ;  how  will  you  find  the  interest  ?  I  give  you  the  time,  the 
rate,  and  the  interest ;  how  will  you  find  the  principal  ?  I  give 
you  the  principal,  the  time,  and  the  interest  jr  how  will  you  find 
the  rate  ?  I  give  you  the  principal,  the  rate,  and  the  interest ; 
how  will  you  find  the  time  ?  I  give  you  the  time,  rate,  and 
amount :  how  will  you  find  the  principal  ? 

5 108.  Wherein  does  discount  differ  from  the  present  worth  ? 

§ 1O9.  What  is  compound  interest  ?  Must  interest  always  be 
added  annually  to  be  compounded  ? 

§  11O.  What  is  a  Bank  ?  How  is  it  established  ?  Relate  all 
you  know  about  it.  What  is  specie  ?  What  are  the  officers  of  a 
bank  called,  and  what  are  their  duties?  Tell  all  you  can  about 
notes  of  hand ;  why  they  are  given ;  by  whom,  &c.  What  is  the 
meaning  of  negotiable  ?  Under  what  circumstances  are  notes  of 
hand  negotiable  ?  What  is  the  amount  called  for  which  the  note 
is  given  ?  What  do  you  understand  by  the  proceeds  of  a  note  ? 

§  111.  What  is  the  precise  difference  between  bank  discount 
and  discount  ?  Which  is  the  more  equitable  ?  Why  ?  When 
the  rate  per  cent,  is  not  specified  in  a  note,  what  rate  is  under- 
stood ? 

§  112.  If  I  give  you  the  present  worth  of  a  bankable  note,  the 
time  and  rate,  how  will  you  find  the  face  of  the  note  ?  Must  in- 
terest on  a  bankable  note  be  computed  for  the  exact  time  specified 
in  the  note  ? 

ANSWERS. 

$97.  (1-8.)  0-02;  0-08;  0'12  ;  0'5  ;  1*06;  1-4;  2-6;  18. 
(9-17.)  0-09;  0-24;  0'99;  1-04;  6-07;  72-81;  0'005 ;  0-025  ; 
0-35.  (18-25.)  0-005;  0-0025;  0'0066f  ;  0'009  ;  0-00375 ; 
0-008331 ;  0'0033| ;  0-004.  (26-32.)  0-025  ;  0-0333^  ;  0'058  ; 
0-36142857-h  2-2025;  5'0'0375 ;  9-4675.  (33-37.)  $56-10  ; 
$4768-08;  $623'875  ;  $2765-40984;  $313111-8945.  (38-47.) 


CHAP.  XII.]  APPENDIX.  34/1 

$4938-17185;  $24690-85925;  $2469-085925;  $32098-117025; 
$3703-6288875;  $62961-6910875;  $246908-5925;  $4069053-- 
6044;  $18765053-03;  $424682-7791.  (48-53.)  1'52  ;  15'52; 
35;  239-4;  0-9125;  0-345.  (54.)  173-67  pounds.  (55.) 
$3-37635.  (56.)  164'15.  (5T.)  $148-50.  (58.)  $46-62  for 
calicoes  ;  $33-30  for  thread  ;  $99'90  for  silks  ;  $42-18  for  broad- 
cloths. (59.)  125  barrels  the  first  time,  75  the  second  time,  and 
300  remaining.  (6O.)  81  per  cent.  (61.)  13*841  bib.  =  \3lb. 
IQoz.  Ipwl.  23^gr.  (62.)  After  the  1st  stroke  90  percent.; 
after  the  2d,  81  per  cent.  ;  after  the  3d,  72T°7  per  cent.;  after  the 
4th,  65TVo  per  ce.it.  (63.)  $6'50.  (64.)  $1620.  (65.)  $200. 
(66.)  40-32o-^.  (67.)  $79725-508. 


§99.  (68.)  $262-50.  (69.)  $405.  (7©.)  $144-585. 
$103-50.  (72.)  $15802-50.  (73.)  $3500.  (74.)  200  bales. 
(75.)  $14750.  (76.)  $8800.  (77.)  $14113-71.  (78-81.) 
$2316-4375;  $2640'5275;  $4370-1645  ;  $9327-1293.  (82-84.) 
$3900;  $5070;  $6591.  (§5.)  $7993-1578+.  (§6.)  $902-50. 
(§7.)  48  per  cent.  (§8.)  $89-91.  (§9.)  $420.  (DO.)  88  trees. 

510O.  (91.)  $6-375.  (92.)  $12-93.  (93.)  $21.  (04.) 
$5-625.  (95.)  $28-125.  (96.)  $  6'50.  (97.)  $1360. 
(98.)  $9863-473.  (99-1OO.)  $5528543-0658;  $2064877-5306. 
(101-103.)  $21;  $75-90;  $3.  (1O4.)  $5203-0625.  (1O50 
$3617-5788. 

§  101.  (106.)  $318-40704.  (1O7.)  $346.  (1O8.)  $316'- 
7262.  (109.)  $2162-8464.  (11O.)  $8-876955.  (111.) 
$31-80.  (112.)  $57-9963.  (113.)  $1020.  (114.)  $4692-60. 
(115.)  $78-3275.  (116.)  $132.  (117.)  $1276'80.  (118.) 
$685.  (119.)  $114-66.  (12O.)  $188-16.  (121.)  $78-40. 
(122.)  $107-016.  (123.)  $33-07425.  (124.)  $74-088. 
(125.)  $240-2355.  (126.)  $874-875.  (127.)  $30.  (128.) 
$2572-50.  (129.)  $249-704.  (ISO.)  2465'75. 

$102.  (131.)  £7-38.  (132.)  $240.  (133.)  $19-6125. 
(134.)  $207-402.  (135.)  $1043-325.  (136-14O.)  $150; 

29* 


APPENDIX.  [CHAP.  xn. 

$225;  $318-75;  $375;  $412*50.  (141-142.)  $12-60;  $13-44. 
(143.)  $85.  (144.)  $136-50.  (145.)  $169.  (146.)  $368. 
(147.)  $82-50.  (148-15O.)  $240;  $360;  $450. 

5 103.  (151.)  $375.  (152.)  $1440.  (153.)  $78-125. 
(154.)  14f  per  cent.  (155.)  10  per  cent.  (156.)  $0-14. 
(157.)  About  34  per  cent.  (158.)  $4-37^.  (159.)  $2'625. 
(160.)  $1120.  (161.)  20  per  cent.  (162.)  25  per  cent. 
(163-164.)  20  per  cent. ;  25  percent.  (165.)  46/j  percent. 
(166.)  31  fi  per  cent.  (167.)  $4000.  (16§.)  $1000. 
(169.)  $100.  (17O.)  9T<V33  per  cent.  (171.)  $4410.  (172.) 
$2284-375.  (173.)  $1102-50.  (174.)  $902-50.  (175- 
176.)  £2  3s.  Od.  3^ far. ;  99  percent.  (177-178.)  $3517*50  ; 
llf  per  cent.  (179.)  $2000.  (ISO.)  $1736'84TV 

§104.  (181-185.)  $8-10;  $29-40  ;  $83-85  ;  $9936-003  J 
$2087487-0105.  (186-19O.)  $61-88;  $129-22;  $898-5431; 
$595323-729 ;  $8565702-60637.  (191-197.)  $6958-257  ; 
$8349-9084;  $9741-5598;  $50099-4504;  $62624-313;  $93936-- 
4695;  $150298-3512.  (198-203.)  $7064-7624;  $11774-604; 
$15699-472;  $21979-2608;  $27317-08128;  $32380-161.  (2O4- 
21O.)  $226-875;  $268-125;  $309-375;  $350'625 ;  $391*875; 
$474-375;  $556-875.  (211.)  $3-332475.  (212.)  $6390'51. 
(213.)  $143-786295.  (214.)  $100.  (215.)  $151-402185. 
(216.)  $0-002747.  (217.)  $0'2625.  (218.)  $89-969925. 
(219.)  $12-53968.  (22O.)  $99'412f.  (221.)  $4*50205. 
(222-224.)  $4998-75;  $5332;  $5235'90.  (225-227.) 
$381-85;  $534-59;  $409-125.  (228-23O.)  $210 ;  $181 ;  $260. 
(231.)  $2-200339}.  (232.)  $92-75.  (233.)  $1-73272. 
(234.)  $91-575.  (235.)  $183-55.  (236.)  $49-2005.  (237.) 
$0-36153.  (238.)  $0-7518825.  (239.)  $89-65.  (24O.)  $85-75. 
(241.)  £15  10s.  3^.  (242.)  £4  10s.  7TW.,  nearly. 
(243.)  £0  3s.  l\d.  (244.)  £4  3s.  6d.  1-95/ar.  (245-247.) 
$507-695|;  $508-535;  $509'374£.  (248-250.)  $407-729145625; 
$408-58990375;  $408-3986241  f.  (251-254.)  $3-993465; 
$63-70948;  $1000-1755+;  $2790-0593+.  (255-257.)  $2-- 
1169+ ;  $2-8729+;  $4'38505+.  (258^-259.)  $5'621175; 


CHAP.  XII.]  APPENDIX.  343 

$111-941685.  (26O-262.)  $3-57;  $761-626576f  ;  $65185-767+. 
(263-265.)  $40-421+;  $80-842+  ;  $29748-866+.  (266.) 
$4-94395.  (267.)  $4-9856125.  (268.)  $1-28646+.  (269- 
271.)  $4984-541f  ;  $5382-375;  $5745-85.  (272-275.) 
$344-4979+;  $348-409+  ;  $356-2313+.;  $360-1423+. 

§  105.  (276.)  $413-79.  (277.)  $5'047.  (278.)  $7-871. 
(279.)  $1-703.  (28O.)  $10'356.  (281.)  $20-88.  (282.) 
$1-75.  (283-284.)  $8-601+;  $7-289+.  (285-287.)  $6-415+; 
$2-742+;  $3-134+.  (288-29O.)  $0'996+;  $M95+;  $1'395+. 

$106.  (291.)  $251-62.  (292.)  $252-12.  (293.)  $5*844. 
(294.)  $5-80.  (295.)  $60-866.  (296.)  $60-72.  (297.) 
$144-404.  (298.)  $143'55. 


(299.)  $29-03.  (SCO.)  $100.  (3O1.)  $40. 
(302.)  $530.  (303.)  $4090'909.  (304.)  $79500.  (3O5.) 
$14133-331.  (306.)  $39328-57j.  (3O7.)  41  per  cent.  (3O8.) 
,5  per  cent.  (3O9.)  7  per  cent.  (31O.)  8  "per  cent.  (311.) 
9  per  cent.  (312.)  5mo.  I2da.  (313.)  lyr.  9mo.  (314.) 
16yr.  8mo.  (315-329.)  20yr.  ;  16fyr.  ;  14f  yr.  ;  12l?/r.; 
lljyr.  :  lOyr.  ;  9TY?/r.  :  8Jyr.  ;  18T2T?/r.  ;  15  Ayr.  ;  13jyr.  ;  ll&yr.  ; 
10|Jyr.  ;  9i}yr.  ;  8|fi/r.  (33O-333.)  3yr.  5mo.  12da.  nearly  ; 
2?/r.  9mo.  3rfa.+  ;  2?/r.  3/wo.  18rf«.  nearly;  lyr.  llmo.  QQda. 
nearly.  (334-^336.)  43?/r.  7wo.  26rfe,+  ;  31yr.  2mo.  4rfa.+  ; 
26?/r.  lOmo.  12^a.  nearly.  (337.)  $530.  (338.)  $4070. 
(339.)  $5-37.  (34O.)  $1437-227.  (341.)  $1437-126.  (342.) 
$49-14.  (343.)  $2970-011.  (344.)  $1165-21+.  (345.) 
$10269-146+. 

§  108.  (346.)  $2-913.  (347.)  $37-411.  (348.)  $2-652. 
(349.)  $86-713.  (35O.)  $1-265.  (351-353.)  $28-35+; 
$32-92+  ;  $37-44,  (354-356.)  $385'66+  ;  .  $459-47+  ; 
$495-98+.  (357-359.)  $58-58+  ;  $42-38+  ;  $46'46+. 
(360-361.)  $76-27+  ;  $86'77+. 

5109.  (362.)  $112-55+.  (363.)  $135-769.  (364.)  $57-881. 


344  APPENDIX.  [CHAP.  xiu. 

(365.)    $262-477.       (366-368.)     $1913-36+;    $1915-04+; 
$1910-12.     (369-37O.)  $43'75+  ;  $44'04+. 


(371.)  $18-085.  (372-376.)  $4'575 ;  $7'438 ; 
$21-231;  $0-867;  $27-809.  (377.)  $1781-10.  (378.)  $6598-48. 
(379.)  $832-89+.  (38O-383.)  $643-175;  $832*81+ ;  $67'- 
62+;  $4214-398+.  (384-385.)  $7'23I;  $6-879+.  (386- 
389.)  $21-35;  $5'425;  $20'29  nearly;  $5-16  nearly.  (39O.) 
$24-15. 

§  112.  (391.)  $609-446.  (392.)  $50'62.  (393.)  $1043-932. 
(394.)  $73-546.  (395.)  $107*594.  (396.)  $376-483.  (397.) 
$3338-75+.  (398-401.)  $152-36+;  $304-72+;  $457-08+ ; 
$507-87+.  (402-4O3.)  $683-877+  ;  $682-667+.  (4O4.) 
$103-518+.  (4O5-4O7.)  $609-446+;  $408-371+;  $309'438+. 
(4O8-41O.)  $1026-079+;  $1031-46+;  $1036'896+. 

CHAPTER  XIII. 

§ 113.  What  do  you  understand  by  Analysis  ?  Give  an  ex- 
ample. What  is  to  be  first  determined  in  a  question  of  Analysis  ? 

5 114.  What  is  Ratio  ?  How  may  a  question  in  analysis  be 
answered  by  ratio  ?  What  does  this  ratio  show  with  respect  to 
one  of  the  given  quantities  ?  Is  it  always  a  multiplier  ?  What 
care  is  specially  necessary  in  determining  the  ratio  ? 

§  115.  Is  there  a  ratio  between  quantities  of  different  denomi- 
nations ?  How  may  such  quantities  be  reduced  so  as  to  have  a 
ratio  ? 

5 116.  What  is  Practice  ?  Has  the  ratio  employed  in  Practice 
a  r  ••!•«!  ion  to  any  one  of  the  given  quantities,  or  to  something  else  ? 

§  117.  What  do  you  understand  by  Currency  ?  What  is  the 
currency  of  the  State  in  which  you  live  ?  What  values  have 
coins  ?  Explain.  How  can  the  state  of  trade  affect  the  value  of 
coin  ?  What  is  the  legal  value  of  the  sovereign  ? 


1HAP.  XIII.]  APPENDIX.  345 

§  § 118j  119.    What  is  one  method  of  reducing  Sterling  to 
Federal  Money  ?     Federal  to  Sterling  Money  ? 
> 

5 120.  What  is  the  method  of  reduction  of  currencies  by 
ratio  ?  What  is  the  ratio  in  this  case  ?  Is  it  fixed  or  variable  ? 
Why? 

§  121.  Give  the  ratios  of  the  dollar  in  the  several  States  to  the 
pound  sterling ;  of  the  pound  sterling  to  the  dollar.  How  is  Fed- 
eral Money  to  be  converted  into  State  or  Canada  currency  ?  How 
these  latter  currencies  into  Federal  Money  ? 

§  122.  What  are  the  custom-house  values  of  some  of  the  chief 
current  foreign  coins  ?  How  may  foreign  coins  be  reduced  to 
Federal  Money  ?  Federal  Money  to  foreign  coins  ? 

ANSWERS. 

§  113.  (1-9.)  36  days;  24  days;  18  days;  9  days;  8  days; 
6  days  ;  4  days  ;  3  days  ;  2  days.  (1O-16.)  $221 ;  $27  ;  $30  ; 
$371;  $45;  $75;  $150.  (17-22.)  40  days;  30  days;  24 
days;  20  days;  12  days;  4  days.  (23-28.)  120  hands;  80 
hands;  40  hands  ;  24  hands;  20  hands;  15  hands.  (29.)  384 
times.  (30.)  $441.  (31.)  $130680.  (32.)  $948*.  (33.) 
$26133331  (34-40.)  $18;  $24;  $30;  $42;  $54;  $66;  $78. 
(41-44.)  $3  ;  $3  ;  $1  ;  $2.  (45-47.)  16  men ;  12  men  ;  8 
men.  (48-5O.)  6f  miles;  Smiles;  10  miles. 

§115.  (51-55.)   f;   J;  ±',   ±;  TirW      (56-6O.)   ^; 

A;  A-J^JTif*.    (6*0  &•    («a-«5.)  1;  -v4-;  tf; 

if-  (66.)  rfA*.  (67.)  T|fs.  (680  Tffr.  (69.)  ^. 
(TO.)  T17.  (71.)  T3^.  (72.)  UftJ.  (73.)  Silver  and  cop- 
per each  T^  of  the  gold ;  silver  and  copper  together  l  of  the  gold. 
(74.)  f  (75.)  ^Wy  (76.)  |4J.  (77.)  JftV.  (78.)  |J}. 
(79-83.)  $1755;  $450;  $675;  $1080;  $1485.  (84-88.) 
$210;  $450;  $1350;  $2160;  $3510.  (89.)  $1290.  (9O.) 
$537f.  (91.)  {  of  a  week.  (92-94.)  30  men;  300  men; 


346  APPENDIX.  [CHAP.  xm. 

18000  men.  (95.)  80  feet,  (96.)  270  feet.  (97.)  941  feet. 
(9§.)  $221iffJ.  (99.)  T5//T  of  a  day.  (1OO-1O8.)  $0'125; 
$0-25;  $0-4375;  $0'541f;  $0'708|;  $0'9375 ;  $M25.  (1O7- 
112.)  4  barrels;  9||  barrels;  16  barrels;  lOf  bushels;  63 
bushels;  105  bushels."  (113-121.)  33  feet ;  66  feet;  99  feet ; 
247|  feet;  330  feet;  4121  feet;  20  rods;  4  rods;  8  rods. 

15   men;    10  men;    6  men;    25  men;  5  men. 

$1-75;  $2'91f;  $4'08£  ;  $6-41§ ;  $9-91f  ;  $21  ; 
$35;  $49;  $142-85$;  $257'14f  ;  $342-85f  (138-141.) 
$56000  ;  $28000  ;  $21000  ;  $35000.  (142-15O.)  9sec. ;  15sec. ; 
45sec. ;  25;  65;  250;  375;  85;  185. 

£  116.  (151.)  $326-25.  (152.)  $4'43i.  (153.)  $20'625. 
(154.)  $3-1625.  (155.)  $11-26125.  (156.)  $6-015jj,  (157.) 
$1-50.  (158.)  $11-81$.  (159.)  $9.  (16O.)  $24'28{. 
(161.)  $7-81|.  (162.)  $6-75.  (163.)  $154'37l.  (164.) 
$68-621.  (165.)  $2-79j.  (166.)  $8'66§.  (167.)  $3'93|. 
(168.)  $121-871.  (169.)  $61'87i  (17O.)  $3-36}.  (171.) 
$5-16f.  (172.)"  IS/o  miles-  (1^3.)  $2'52|.  (174.)  $10. 
(175-18O.)  720  bushels;  1440  bushels;  2880  bushels;  1080 
bushels;  1800  bushels;  2160  bushels.  (181-185.)  192  beg- 
gars; 120  beggars;  96  beggars;  72  beggars;  60  beggars. 
(186-190.)  $150;  $25;  $16'66|;  $8-33i ;  $1-66|.  (191- 
195.)  266fyd. ;  240z/d;  160^.;  145T5T?/£/.;  nS^yd.  (196- 
20O.)  39666|  miles;  51000  miles;  62333^  miles;  107666f 
miles  ;  147333^  miles. 

5  122.  (201-21O.)  £1  ;  £4  ls.+  ;  £6  14s.  2d.  2|/ar.+  ; 
JC12  3s.  9d.  2l/ar.  nearly;  £181  Is.  lOd.  l^/«r.+  ;  £5"l2s.  3d. 
3/ar.+  ;  £261  4s.  Id.  l|/ar.+;  £4565  6s.  10^.  3/ar.+;  £92352 
17s.  2d.  l/«r.+  ;  £572089  8s.  Bd.0far.+  .  (211-218.)  £l 
16s.;  £5  3s.  Id.  3/ar.-^  ;  £16  8s.  Id.  2far.+ ;  £147  Is.  3d. 
2far.+  ;  £1794  14s.  Id.  3far.+  ;  £16477  5s.+  ;  £18458  9s.  6d. 
"2far.+  ;  £44700  11s.  7d.  lfar.+.  (219-234.)  £l  15s.  8d.+ ; 
£5  2s.  2d.  2/ar.+  ;  £16  5s.  2d.+ ;  £145  14s.  6d.  3/ar.+  ;  £1778 
7s.  10d.+  ;  £16327  9s.  Id,  3/«r.+  ;  £18290  13s.  5d.  2far.+ ; 
£44294  4s.  2d.  3far.+  ;  £l  16s.  Id.  3far.+  ;  £5  3s.  7d.  2far.+  ; 


CHAP.  XIII.]  APPENDIX.  347 

£16  95.  <7d.  S/ar.-f ;  £147  14s.  Wd.  lfar.+  ;  £1802  19s.  6rf. 
2/ar.+ ;  £16553  3s.  7dL  3/ar.+ ;  £18543  10s.  9rf.  l/ar.-f ; 
£44906  11s.  5rf.  3/ar.+.  (235-24O.)  $44'77 ;  $131-50+; 
$188-94+  ;  $1310-379+;  $20234-265+;  $336283'38+.  (241- 
258.)  $44-40;  $130'41+  ;  $187-38  ;  $1299-55  nearly;  $20067'- 
039+ ;  $333504-144;  $44-811+ ;  $131-622+;  $189-115; 
$1311-582+;  $20252-845+;  $336592-188+;  $46'22f;  $132'- 
829+;  $190-85+;  $1323-615+;  $20438-651+;  $339680'17i. 
(259-263.)  £25  Is.  Canada;  £23  7s.  7Jrf.  Georgia;  £30  Is. 
2§J.  New  England;  £37  11s.  6d.  Pennsylvania;  £40  Is.  7^. 
New  York.  (264-26§.)  £9  6s.  W-Qd.  Canada;  £8  14s.  4-72d. 
Georgia;  £11  4s.  2'64d.  New  England;  £14  Os.  3'3d.  Pennsyl- 
vania; £14  18s.  ll-52<2.  New  York.  (269-273.)  £250  Can- 
ada; £233  6s.  8d.  Georgia;  £300  New  England;  £375  Penn- 
sylvania; £400  New  York.  (274-278.)  $303-10  Canada; 
$324-75  Georgia;  $252'58i  New  England;  $202'06|  Pennsyl- 
vania; $189-43|  New  York.  (279-283.)  $321-05  Canada; 
$343-982 \  Georgia;  $267'541|  New  England ;  $214'03i  Penn- 
sylvania; $200-656^  New  York.  (284-288.)  $4000  Canada; 
$4285-714f  Georgia;  $3333-33^  New  England;  $2666'66f  Penn- 
sylvania; $2500  New  York.  (289.)  1755-^  sovereigns. 
(290.)  11575f|  five  francs.  (291-295.)  595TW  Mexican 
doubloons;  1190|f  ten  thalers;  £2321  2s.  4^.  Canada ;  18568-94 
rupees  of  Bengal ;  11605-5875  ducats  of  Naples.  (296-301.) 
$5904-675;  $12740;  $3659-625;  $385660-88;  $1342851*78; 
$369698-994.  (302.)  $84760'33£.  (3O3.)  £5113  15s.  9d. 
2/ar.+.  (3O4.)  £1008  12s.  IQd.  lfar.+  New  England.  (305.) 
26455^|  five  francs.  (3O6.)  $0*003388.  (3O7.)  $4-356. 
(3O8.)  $29-04.  (3O9-314.)  416f  ounces  of  Sicily;  1250 
ducats  of  Naples ;  2061  ffi  florins  of  Augsburg;  1269|f  rix  dol- 
lars of  Bremen ;  6434g-  Mexican  doubloons;  219^f  Louis-d'ors. 
(315-325.)  $15600;  14716|f  crowns;  3223ryT  sovereigns; 
14857|  specie-dollars  of  Denmark;  14716f|  specie-dollars  of 
Norway;  8472|§  pagodas  of  India;  31200  rupees  of  Bengal; 
13866f  milrees  of  Portugal ;  44571^  mark  bancos  of  Hamburg; 
3120  English  guineas;  83870§f  francs.  • 


348  APPENDIX.  [CHAP.  xiv. 


CHAPTER  XIV. 

§  123.  Explain  a  proportion.  What  names  are  given  to  the 
terms  of  a  ratio  ?  What  of  a  proportion  ?  Wherein  does  a  ratio 
differ  from  a  proportion  ?  Whence  did  the  Rule  of  Three  derive 
its  name  ?  What  is  meant  by  known  quantities  ?  What  by  the 
unknown  ?  What  is  a  mean  proportional  ?  How  is  it  found  ? 
What  propositions  are  true  with  respect  to  the  various  terms  of  a 
proportion  ?  Repeat  the  first  form  of  the  Rule  of  Three.  Ex- 
plain its  principles.  Repeat  the  second  form  of  this  rule.  Ex- 
plain the  difference  between  the  two  rules.  Show  the  connection 
between  the  second  form  and  the  method  by  analysis. 

§  124.  Explain  a  compound  proportion.  By  how  many  pro- 
cesses may  a  question  involving  complex  conditions  be  answered  ? 
Illustrate  by  an  example. 

125.  What  is  Arbitration  of  Exchange?     What  principle 
does  it  involve  ? 

•  126.  What  is  Partnership?  What  principle  lies  at  the 
foundation  of  operations  under  this  head  ?  How  will  you  ascer- 
tain each  partner's  gain  or  loss  ? 

§  127  Wherein  does  Double  Fellowship  differ  from  partner- 
ship? 

ANSWERS. 

§  123.  (1.)  §40-56.  (2.)  $21.  (3.)  £3  16s.  6d.  (4.)  52j 
weeks.  (5.)  113£  miles.  (6.)  14f  years.  (7.)  320  yards. 
(8.)  25  men.  (9.)  2000  pounds.  (1O.)  61f  bushels.  (11.) 
11  seconds.  (12.)  3600  times.  (13.)  31|  minutes.  (14.) 
8£  days.  (15.)  13014f  pounds.  (16.)  66JH  feet.  (17.) 
6js.  (1§.)  $104-16f.  (19.)  23£  miles.  (2O.)  4  hours. 
(21.)  $576.  (22.)  $11.  (23.)  $13.  (24.)  $9510.  (25.) 
$1732-25.  (26.)  $310.  (27.)  550  bushels.  (2§.)  12|  bar- 


CHAP.  XIV.]  APPENDIX.  349 

rels.  (29.)  33£yd.  (30.)  32  miles.  (31.)  $42-85?.  (32.) 
Uda.  4hr.  (33.)  9|  inches.  (34.)  8  days.  (35.)  45g 
yards.  (36.)  $100.  (37.)  $14400.  (38.)  $1166|.  (39.) 
68104T££T  miles.  (4O.)  1037^  miles.  (41.)  60590592000000 
miles.  (42.)  6hr.  40mm.  (43.)  $625.  (44.)  H6da.  4$hr. 
(45.)  _!_.  of  an  inch.  (46.)  $4-66f.  (47.)  $5-25.  (48.) 
\n\mo.  (49.)  76032  years.  (50.)  60  feet.  (51.)  31  miles. 
(52.)  ISyd.  (53.)  29^  degrees.  (54.)  236|  miles  with  cur- 
rent, and  168|  miles  against  it.  (55.)  ^f  J}T  times.  (56.)  |f. 
(57.)  Supply-pipe  alone  8%da.  ;  supply-pipe  and  1st  dischargirg 
pipe  \Q\da.  ;  supply-pipe  and  2d  discharging  pipe  llfda.  ;  all  to- 
gether 15da. 


(58.)  441T3T  miles.  (59.)  71  Ij  pounds.  (6O.)  69 
weeks.  (61.)  30  men.  (62.)  3291^.  (63.)  6  men.  (64.) 
11  men.  (65.)  30  pounds.  (66.)  $144.  (67.)  500  men. 
(68.)  288^  days.  (69.)  200  men.  (70.)  $67'50.  (71.) 
$14-40.  (72.)  8  months.  (73.)  6  per  cent.  (74.)  $360. 
(75.)  $19-20.  (76.)  80  cows.  (77.)  32yd.  (78.)  8mo. 
(79.)  10  persons.  (8O.)  $50  dollars'  worth. 

$125.  (81-83.)  $1899-3911;  $1816'66f;  $83-73  j.  (84- 
86.)  $5398-65;  $4755'55|;  $643'09^.  (87.)  36|£}  pounds. 
(88.)  375  pears.  (89.)  20  £ff  English  guineas.  (9O.)  982^ 
francs. 


(91.)  A.'s  share  $100-80,  B.'s  $109*20.  (92-94.) 
A.  $200  ;  B.  $222-22f  ;  and  $0«55f  on  the  dollar.  (95.)  $138'- 
46T2T,  $161-531^.  (96.)  $11-25,  $18-75.  (97.)  $400,  $600, 
$900,  $208-33i,  $291-66f.  (98.)  $11-50,  $5'75,  $9-20.  (99.) 
$161,  $112,  $92.  (10O.)$342-85f,$285-71f,  $171-42f.  (1O1.) 
$240,  $120,  $80,  $60. 

5127.  (1O2.)  $54-92ff,  $45-07^-.  (1O3.)  $55'32£f, 
$94-67£f  (104.)  $3-50.  (1O5.)  $39'62jf,  $60'37f§.  (1O6.) 
A.  $17-50,  B.  $4-65,  C.  $10-60,  D.  $5-75,  E.  $8-00.  (1O7.)  A. 
$13-50,  B.  $12-00,  C.  $30-00.  (1O8.)  A.  $195,  B.  $112-50,  C. 

30 


35C  APPENDIX.  [CHAP.  xv. 

$67-50      (1O9.)  A.  $40,  B.  $30,  C.  $24.     (11O.)  Each  officer 
,  each  midshipman  $80,  each  sailor  $30. 


CHAPTER  XV. 

§  128.  How  do  you  find  the  average  of  a  series  of  numbers  ? 
Define  average.  To  what  topics  is  the  principle  of  average  ap- 
plied under  this  chapter  ? 

§  5  129,  130.  What  is  Equation  of  Payments  ?  Give  the 
rule,  and  illustrate  it 

§  131.  What  is  the  rule  for  finding  the  cash  balance  ?  Ex- 
plain the  principle  involved  in  the  rule.  If  the  cents  of  an  entry 
exceed  50,  what  may  be  done  ?  if  less  than  50,  what  ? 

5  §  132,  133.  Define  Alligation  Medial.  In  what  does  it  re- 
semble equation  of  payments?  What  is  Alligation  Alternate? 
Show  in  what  way  Alligation  may  be  of  use  to  the  dealer  in  gro- 
ceries. 

ANSWERS. 

5128.  (1-5.)  SJ;  4;  5;  6;  7.  (6-1O.)  6};  16$;  16J; 
69-J  ;  7|.  (11.)  $1220-181.  (12.)  93TV  (13.)  Average 
leeAft.  ;  aggregate  1163^&.  (14.)  £5  16s.  3d.  ±qr.  (15.) 
36yr.  6mo.  l$da.  (16.)  29-67  inches.  (1T-1§.)  22°44'44f"; 
5m.  7s.  (19.)  39-09281  inches.  (2O.)  14-5051475.  (21- 
22.)  45m.42is.  ;  2m.  41T3T6irs-  (23.)  6oz.  18±pwt.  (24-25.) 
29°;  42i°. 


(26.)  4TV?w.=4mo.  16da.  (2T.)  8fmo.=8mo. 
(28.)  9mo.  (29.)  5fjmo.=5mo.  22<fo.  (SO.)  6mo.  (31.) 
6^mo.  (32.)  64TV  days  after  Jan.  1st,  or  March  5th.  (33.)  96| 
after  the  1st  of  March,  or,  calling  it  97  days,  we  have  June  6th. 
(34.)  6^mo.=6mo.  12da.  nearly.  (35.)  16^|  days  after  the 
10th  of  March,  or  the  27th  of  March.  (36.)  5^  months.  (37- 


CHAP.  XVI.]  APPENDIX.  351 

38.)  4fmo.  ;  4±mo.     (39.)  28  f£  days  after  July  1st;  that  is,  on 
the  30th  of  July. 

§130.  (4O.)  117|  days.  (41.)  4|  months.  (42.)  2J 
months.  (43-44.)  $875;  8£rao. 

$131.  (45.)  $148-51  in  favor  of  A.  (46-47.)  Cash  bal- 
ance of  $229-26  in  favor  of  B.  ;  cash  balance  of  $232-72  in  favor 
of  B.  (48.)  Cash  balance  of  $98'92  in  favor  of  A.  (49-5O.) 
Cash  balance  of  $174*56  in  favor  of  A.  ;  cash  balance  of  $180*28 
in  favor  of  A. 


§  132.    (51.)  $0-502.     (52.^  $0-83}i.     (53.)    12f£  ,arats. 
(54.)  67  ji  degrees.    (55.)  6f  knots.    (56.)  11T5T  cents.    (57.) 

2239-gCf.      58.)  $3-86|f.     (59.)  83^lb.     (6O.) 


§  133.  (61-66.)  12,  52  ;  17,  47  ;  24,  40  ;  36,  28  ;  43,  21  ; 
54,  10.  (67.)  Twice  as  much  wine  as  water.  (68-71.)  4,  4, 
4,  9  ;  3,  3,  3,  12  ;  2,  2,  2,  15  ;  1,  1,  1,  18.  (72-76.)  17,  4,  4  ; 
10,  10,  12  ;  7,  7,  18  ;  5,  5,  22  ;  3,  3,  26.  (77-81.)  5,  5,  5,  6  ; 
4,  4,  4,  9;  3,  3,  3,  12  ;  2,  2,  2,  15;  1,  1,  1,  18.  (82-85.)  90, 
90,  810  ;  90,  90,  405  ;  10,  10,  10  ;  4f,  4f,  4f  .  (86-91.)  62 
pounds  of  each  ;  39  pounds  of  each  ;  25^  pounds  of  each  ;  16 
pounds  of  each  ;  9|  pounds  of  each  ;  4|  pounds  of  each.  (92- 
96.)  ~100,  100,  2100  ;  100,  100,  900  ;  "lOO,  100,  500  ;  100,  100, 
300;  100,  100,  180. 


CHAPTER  XVI. 

§  134.  What  is  Involution  ?  What  is  a  square  ?  a  cube  ? 
What  is  the  difference  between  a  root  and  a  product  ?  between 
the  power  of  a  number  and  the  factor  of  a  number  ? 

§  135.  What  is  meant  by  Evolution  ?  What  is  the  difference 
between  the  4th  power  of  a  number  and  the  4th  root  of  a  number  ? 

§  136.  To  what  is  the  square  of  two  numbers  (say  tens  and 
units)  equal  ?  the  square  of  three  numbers  ? 


352  APPENDIX.  [CHAP,  xvi 

§  137.  Explain  how  the  number  of  figures  in  a  square  root 
may  be  determined.  Illustrate  by  a  diagram  the  involution  of 
two  or  more  numbers  to  the  2d  power. 

5 138.  Illustrate  the  extraction  of  the  square  root  by  a  geomet- 
rical diagram.  What  is  the  rule  for  the  extraction  of  square  root  ? 
Why  must  a  naught  be  annexed  to  the  trial  divisor  ?  If  there 
are  decimals,  how  must  the  number  be  pointed  off?  How  is  the 
square  root  of  a  common  fraction  found  ? 

§ 139.  What  is  a  triangle  ?  What  is  a  right-angled  triangle  ? 
What  is  an  hypotenuse  ?  Give  the  established  proposition  in  ge- 
ometry concerning  the  square  of  the  hypotenuse.  What  relation 
do  similar  surfaces  or  areas  bear  to  each  other  ? 

5 140.  The  cube  of  two  numbers  equals  what  ?  of  the  sum  of 

any  number  of  numbers  ? 

§  §  141,  142.  On  what  principle  may  the  number  of  figures 
in  the  root  of  a  cube  be  determined  ?  Illustrate  by  a  diagram  the 
involution  of  two  numbers  to  the  3d  power.  Illustrate  by  dia- 
gram the  extraction  of  the  cube  root.  Give  the  rule.  Why  must 
two  naughts  be  annexed  to  the  trial  divisor  ?  To  what  extent 
may  the  extraction  of  a  root  be  illustrated  geometrically  ?  How 
is  a  decimal  to  be  pointed  off?  How  is  the  cube  root  of  a  deci- 
mal to  be  found  ?  of  a  common  fraction  ? 

§  143.  Similar  solids  are  to  each  other  as  what  ?  Illustrate. 
Can  you  give  an  example  not  found  in  the  book  of  the  use  of  a 
knowledge  of  square  or  cube  roots  ? 

ANSWERS. 

§134.  (1-10.)  196;  361;  576;  1296;  2304;  3249;  8649; 
12321;  28224;  54289.  (11-2O.)  2197 ;  5832;  12167;  42875  ; 
85184;  175616;  753571;  3241792;  37933056;  10793861. 
(21-22.)  229345007;  387420489.  (23-26.)  0'5625;  1-2996; 
1162-1481;  22-857961.  (27-31.)  0'216;  0-002197;  0-00824- 
•2408;  0-274625;  27-570978261.  (32-36.)  5TV  5  13£ 


CHAP  XVI.]  APPENDIX.  353 


T.     (37-41.)   J;    A;    38TJf  r  ;    172fff; 
(42-44.)  l57-&823y;  0-00390625;  0-019775390625. 

$138.  45.)  104976.  (40.)  526112.  (47-52.)  Ill;  232; 
5555;  130321;  923521;  248832.  (53.)  59'049.  (54-57.) 
2-56;  0-0625;  4-123  nearly;  6-123  nearly.  (58.)  O'OOlll. 
(59-60.)  f-  ;  |f.  (61-62.)  $;  2],  (63-65.)  2-027  nearly; 
0-8044  nearly  ;  0'515+. 

§139.  (66.)  80  feet.  (67.)  94-34  miles  nearly.  (68.) 
4-24264  feet  nearly.  (69.)  20  feet.  (7O.)  5'196  feet  nearly. 
(71.)  25  times.  (72.)  36  gallons.  (73.)  3-60555  feet  nearly. 
(74-77.)  20  feet  ;  18-86796+  feet  ;  15-54723+  feet  ;  22-36067+ 
feet.  (78-8O.)  672  feet;  680  feet;  104  feet. 


.  (§1-9O.)  703+3x702x5+3X70X52+53;  803+3X 
80*X9+3X80X92+93;  1003+3X  1002X40+3X  100X40a+ 
403+3  X  (1  00+40)2  X  2+3  X  (100+40)  X  22+23  ;  3003+3  X  300aX 
60+3  X  300  X602+608+3X  (300+60)a  X  5+3  X  (300+60)  X52+ 
5s;  403+3X402x7+3X40X72+73;  903+3X902X6+3X90x 
62+63  ;  2003+3X2002X20+3X200X20a+203+3X(200+20)2X 
1+3X(200+20)X12+13  ;  4003+3X4002X90+3X400X902+ 
903+3X(400+90)2X6+3X(400+90)X62+6S;  8003+3X8002X 
70+3  X  800  X  70a+703+3  X  (800+70)2  X  9+3  X  (800+70)  X  9a+93; 
9003+3X9002X90+3X900X902+903+3X  (900+90)2  X9+3X 
(900+90)  X92+93. 


(91.)  216.  (92-95.)  1331;  35;  49;  1936.  (96.) 
0-7773.  (97.)  2-65.  (98.)  1-08005+.  (99.)  T2599+.  (1OO.) 
2-08008+.  (1O1.)  1-4422+.  (1O2-1O4.)  jf  ;  f£  ;  2-577 
nearly.  (1O5-11O.)  1-726  nearly;  0-9353  nearly";  0-8736 
nearly;  0-9196  nearly  ;  P6631  nearly  ;  2-3996  nearly. 

§143.  (111.)  18ff  pounds.  (112.)  6-1476  inches  nearly. 
(113.)  6  inches.  (114-115.)  The  weight  of  the  first  three  is 
equal  to  the  weight  of  the  last  three  ;  the  diameter  of  a  ball  of 
average  weight  is  30-2381  inches  nearly.  (116.)  50-3968 


354:  APPENDIX.  [CHAP.  xvn. 

inches  nearly  in  length,  and  40*3175  inches  nearly  for  diameter. 
(117.)  47-984  inches  nearly.  (11§.)  64.  (119.)  37-6412 
inches  nearly  in  length,  28-2309  inches  nearly  in  breadth,  and 
18-8206  inches  nearly  in  depth. 


CHAPTER  XVII. 

§  144.  What  is  an  Arithmetical  Progression  ?  What  is  the 
difference  between  an  ascending  and  a  descending  series  ?  What 
five  quantities  are  to  be  considered  in  arithmetical  progression  ? 
How  may  the  last  term  of  a  series  be  found,  if  the  first  term,  the 
common  difference,  and  the  number  of  terms  are  given  ?  Ex- 
plain the  principle.  How  may  the  sum  of  all  the  terms  be  found, 
when  the  first  term,  the  last  term,  and  the  number  of  terms  are 
given  ? 

§  145.  Will  you  explain  the  difference  between  an  arithmetical 
and  a  geometrical  progression  ?  between  the  ascending  and  the 
descending  series  ?  What  five  quantities  are  to  be  considered  ? 
Given  the  first  term,  the  ratio,  and  the  number  of  terms,  how  will 
you  find  the  last  term  ?  How  will  you  find  the  sum  of  the  terms, 
if  first  term,  last  term,  and  ratio  are  given  ?  What  is  the  value 
of  the  last  term  of  an  infinite  series  ? 

ANSWERS. 

§  144.  (1.)  299.  (2.)  25^.  (3.)  $19.  (4.)  3J.  (5.)  12 
inches.  (6.)  167  hills.  (7.)  $80.  (§.)  442.  (9.)  50800. 
(1O.)  1825  miles.  (11.)  £6  6s.  (12.)  560000.  (13.)  146217. 
(14-15.)  78;  300.  (16.)  402^  feet.  '  (17.)  $570. 

§145.  (!§•)  64.  (19.)  327680.  (2O.)  320  miles.  (21- 
23.)  $16000;  $64000;  $256000.  (24.)  118096.  (25.)  436905. 
(26.)  1398101  pecks.  (27.)  U.  (2§.)  ',  (29.)  £.  (3O« 
31.)  20  feet;  10  feet.  (32.)  9.  (33.)  5. 


CHAP.  XVIil.j  APPENDIX.  355 


CHAPTER  XVIII. 

§  146.  What  is  a.  rectangle  ?  How  is  the  area  of  a  rectangle 
found  ?  of  a  parallelogram  ?  of  a  triangle  ?  of  a  trapezoid  ?  the 
circumference  of  a  circle,  iis  diameter  'being  given  ?  the  area  of 
a  circle,  its  diameter  being  known  ?  How  is  the  volume  of  a  prisrn 
or  of  a  cylinder  found  ?  How  is  the  volume  of  a  pyramid  or  of  a 
cone  found  ?  the  volume  of  a  sphere,  its  diameter  being  given  ? 
the  volume  of  the  frustum  of  a  pyramid  or  of  a  cone  ?  How  is 
the  area  of  an  ellipse  found  ? 

ANSWERS. 


(1.)  376s?.  /*.  =41^s?.  yd.  (2.)  2394s?.  in.=l6lsq. 
ft.  (3.)  91s?.  rods=-f^f  of  an  acre.  (4.)  110|s?.  in.  (5.) 
The  rectangular  piece  contains  9s?.  in.  more  than  the  square  one. 
(6.)  54s?.  //.  (7.)  37-  95s?.  ft.  (8.)  84s?.  rods.  (9.)  845?. 
yd.  (1O.)  80s?,  yd.  (II.)  40s?.  rods=i  of  an  acre.  (12.) 
16fs?.  ft.  (13.)  25132-8  miles  nearly.  (14.)  119-38  feet. 
(15.)  219-21  times.  (16.)  502-656  acres.  (17.)  The  first  is 
equal  to  the  sum  of  the  other  two.  (18.)  30  cubic  feet.  (19.) 
14-966  cubic  feet.  (20.)  106-029  cubic  inches.  (21.)  93244729f 
cubic  feet.  (22.)  91-1064  cubic  feet.  (23.)  201061930s?. 
miles.  (24.)  113'0976s?.  in.  (25.)  113-0976  cubic  inches. 
(26.)  904-7808  cubic  inches.  (27.)  537-2136  cubic  inches= 
1-909  beer  gallons.  (28.)  101f|  cubic  feet.  (29.)  4712'4s?. 
ft.  (30.)  2078-1684S?.  in.==14' 


(31.)  $1159-64  nearly.  (32.)  6  months.  (33.) 
1159-42  nearly.  (34.)  10  If}  rods.  (35.)  A.'s  $184T83,  B.'s 
$153}i,  C.'s  $161TV  (36.)  $500.  (37.)  146000  words. 
(38.)  1039rVdays.  (39.)  $128.  (4O.)  $0'96.  (41.)  20  per 
cent.  (42.)  $1-50.  (43.)  In  8  months.  (44.)  $160.  (45.) 
22  men,  18  women,  50  children.  (46.)  $2-91  nearly.  (47.) 
292mm.,  9231^.  (48.)  $584*41  +  .  (49.)  A.'s  $135-70,  B.'s 
$203-55,  C.'s  $271-40.  (5O.)  2000  specie-dollars.  (51.)  $76'52-K 


356  APPENDIX.  [CHAP.  xvm. 

(52.)  9856  years.  (53.)  24856-28+  miles.  (54.)  90  men. 
(55,)  18  days.  (56.)  3fcr.  26r8fmiw.  (57.)  All  could  do  it  in 
16  days;  A.  in  52^  days;  B.  in  57f  J ;  C.  in  44T4g;  D.  in  280 
days.  (58.)  9|  per  cent.;  19}  per  cent.;  28|  per  cent. 
(59.)  2-418115599  gallons.  (6O.)  2521;  5041  ;  7561  ;  10081. 
(61.)  2519;  5039;  7559;  10079.  (62.)  36000  rations. 
(63.)  72  yards.  (64.)  5050  pigeons.  (65.)  120  casks.  (66,) 
40  pounds  of  sugar.  (67.)  f ^  _  i£|3Jj.  Cremona  feet. 
(6§.)  7^  days.  (69.)  lij-  days.  (TO.)  24  ounces  per  day. 
(71.)  A.'s  1428*,  B.'s  $5~71f.  (72.)  A.  paid  $327T3T,  B.  paid 
$163F7T,  C.  paid  $109Ty.  (73.)  120  yards.  (74,)  $328074-37^. 
(75.)  Between  60  and  61.  (76.)  77440  miles.  (77.)  $4000. 
(7§.)  I  neither  gain  nor  lose.  (79.)  I  lose  ^  of  a  cent  on  each 
orange.  (8O.)  I  neither  gain  nor  lose.  (81.)  I  gain  y1^  of  a 
cent  for  each  cent  employed.  (§2.)  2T2^  times.  (83.)  2  times. 
(84.)  15  yards.  (85.)  The  younger  had  $453-846  nearly;  the 
elder  had  $546*154  nearly.  (86.)  A.  has  $30,  B.  $33,  C.  $37. 
(87.)  If  days.  (88.)  Value  of  horse  $150,  number  of  tickets 
60.  (89.)  Each  received  $50,  and  Henry  sold  200  melons. 
(9O.)  48  apples  for  20  cents.  (91.)  He  gained  80  cents. 
(92.)  A.  has  $266|,  B.  has  $933i.  (93.)  $2700.  (94.)  25 
percent.  (95.)  20  per  cent.  (96.)  $2'625.  (97.)  53}  yards. 
(98.)  9ffff|  years.  (99.)  12-907  inches,  nearly.  (1OO.)  300 
strokes.  (1O1.)  At  the  end  of  every  5  hours.  (1O2.)  $2-586. 
(1O3.)  825000.  (104.)  J  of  a  unit.  (1O5.)  A.  has  $133*79-, 
B.  has  $80-27+,  C.  has  $30-10+.  (1O6.)  A.  ought  to  have 
$135-85,  B.  $81-51,  C.  826-80.  (1O7.)  3j/V  days.  (1O8.)  There 
was  due  $2952-28.  (1O9.)  The  deer  weighed  168  pounds. 


The  total  number  of  examples  under  the  different  chapters  is  as 
follows : 

Chap.  I,  0.  Chap.  II.,  92.  Chap.  Ill,  58.  Chap.  IV.,  132. 
Chap.  V.,  262.  Chap.  VI,  320.  Chap.  VIL,  25.  Chap.  VIII.,  190. 
Chap.  IX.,  370.  Chap.  X.,  575.  Chap.  XL,  700.  Chap.  XII,  410. 
Chap.  XIIL,  325.  Chap.  XIV.,  110.  Chap.  XV.,  96.  Chap.  XVI, 
110.  Chap.  XVII,  33.  Chap.  XVIIL,  109.— Total,  3926. 


PERKINS'   COURSE  OF  MATHEMATICS. 

TESTIMONI  ALS. 

From  the  numerous  recommendations  of  these  works,  received  from  the  highest  sources,  the 
following  selections  are  deemed  sufficient  to  call  the  attention  of  Teachers,  and  those  connected 
with  Education,  to  a  thoroughjrial  of  their  merits. 

From  PROP.  COOK  and  DR.  CAMPBELL,  of  Albany  Academy. 

u  From  all  who  have  used  the  ELEMENTARY  ARITHMETIC  here,  both  teachers  and  scholars, 
we  hear  bur  one  opinion,  and  that  is  most  favorable.  It  is  an  excellent  text-book,  and  wo  hira 
QO  hesitation  in  recommending  it  to  parents  and  pupils." 

Extract  from  the  Minutes  of  the  Board  of  School  Commissioners  of  the  City  of  Albany , 

April  11/A,  1850. 

"  The  Committee  on  text-books  made  a  report,  recommending  Perkins'  SERIES  OP  ARITHME- 
TICS as  superior  text-books  for  the  use  of  schools  ;  whereupon  it  was  unanimously  resolved, 
that  Perkins'  Primary  and  Elementary  Arithmetics  be  adopted  as  the  Arithmetical  text-books  ol 
ll»e  Albany  District  Schools." 

From  I.  W.  JACKSON,  A.  M.,  Professor  of  Mathematics,  Union  College. 
'  The  Higher  Arithmetic  is  a  work  of  an  order  superior  to  any  that  has  been  issued  from  try; 
American  press.     Indeed,  I  am  acquainted  with  no  work  on  Arithmetic  in  the  English  anguaga 
equal  to  it.     I  am  confident  that  its  general  adoption  as  a  text-book,  by  our  seminaries,  would 
be  considered  by  all  who  feel  an  interest  in  the  promotion  of  the  exact  sciences,  as  an  omen  of 
good." 
The  following  was  unanimously  adopted  by  the  Chenango  County  Teachers'  Institute,  in 

October,  1S19. 

"That  we  consider  Perkins'  Elementary  Arithmetic,  Higher  Arithmetic  and  Elements  at 
Algebra,  better  adapted  to  the  use  of  schools  than  any  other  works  on  these  subjects :  that  wa 
highly  recommend  their  use  throughout  the  country,  and  that  we  will  each  use  our  endeavors  to 
secure  their  adoption  in  our  several  schools." 

From  SAMUEL  CROSS,  Principal  of  Classical  Institute^  Warren,  R.  I. 
"  After  a  thorough  examination,  I  must  say,  in  all  candor,  that  I  am  much  pleased  with  tlu 
Elementary  Arithmetic." 

From  GEO«GE  W.  MEEKER,  ESQ..,  Secretary  of  the  Board  of  School  Inspectors,  Chicago. 
"  I  consider  them  better  adapted  to  the  wants  of  our  Schools  and  Seminaries  than  any  othef 
series  extant.    I  am  happy  to  add,  that  the  Arithmetics  have  recently  been  unanimously  adopteo 
as  the  text-books  of  the  Public  Schools  of  this  City. 

From  each  of  the  Principals  of  the  Public  Schools  of  Chicago. 
"  We  have  examined  Perkins'  Arithmetics,  and  consider  them  the  best  before  the  public." 

Extract  from  a  Letter  from  GEO.  P.  WILLIAMS,  Professor  of  Mathematics  and  Natural 

Philosophy,  University  of  Michigan. 

"  After  an  examination  of  the  last  editions  of  these  works,  I  am  prepared  to  repeat  the  opink 
formerly  expressed,  that  they  are  the  best  arithmetics  in  use." 

From  C.  M.  WRIGHT,  Principal  of  South  Bend  (Ind.)  Academy. 

u  1  ha  ^e  adopted  Perkins'  Mathematical  Series,  and  say  unhesitatingly  that  they  need  only  I 
crammed  to  be  liked." 

From  JOHN  F.  NICHOLS,  Principal  of  the  High  School,  Detroit. 

"  I  have  examined,  with  great  care,  the  revised  edition  of  Perkins'  Elementary  Arithmetic , 
and  I  l-3sitate  not  in  saying,  that  in  my  opinion,  it  is  much  superior  to  any  other  work  on  th« 
•ubject,  with  which  I  am  acquainted." 

From  O.  S.  TAYLOR,  Principal  of  Branch  University,  Michigan. 
'•The  Elementary  Arithmetic  is  the  most  practical  work  in  the  English  language." 

From  N.  BRITTAIN,  Principal,  of  Rochester  Collegiate  Institute. 

I  have  examined  with  considerable  care  "Perkins'  Elementary  Arithmetic,"  anl  regard  it  a 
ivork  of  great  merit.  In  several  important  points,  I  consider  it  preferable  to  any  work  of  tl.« 
kind  now  in  use. 

From  I.  F.  MACK,  Esq..,  late  Sup.  P.  Schools  of  Rochester. 

I  concur  in  the  sentiment  above  expressed  in  reference  to  •'  Perkins'  Elementary  Arithmetic.' 
The  "  Higher  Arithmetic,"  and  Algebra,  by  the  same  Author,  have  been  adopted  by  the  Board  cil 
Education  of  this  city,  and  introduced  into  the  public  schools  with  entire  satisfaction. 
From  CHARLES  AVERY,  A.  M.,  A.  A.  S.,  Prof,  of  Chemistry  and  Nat.  Phil..  Hamilton  College 

I  have  examined  Perkins'  Algebra,  and  am  pleased  with  it.  I  esteem  it  a  valuable  work  of 
the  kind;  and  do.  therefore,  cheerfully  recommend  it  to  the  confidence  and  patronage  of  (*»' 


A  DICTIONARY  OP  THE  ENGLISH  LANGUAGE, 

*  JONTAININQ     THE    PRONUNCIATION,    ETYMOLOGY,    AND     EXPLANATION    OF    ALL    WORDS     Al> 

THORIZED   BY  EMINENT   WRITERS  ; 

To  which  are  added,  a  Vocabulary  of  the  Roots  of  English  Words,  and  an  Accented 
List  of  Greek,'Latin,  and  Scripture  Proper  Names 

BY  ALEXANDER  REID,  A.M., 

Rector  of  the  Circus  School,  Edinburgh. 

VV:.tn  a  '.'ritical  Preface,  by  HENRY  REED,  Professor  of  English  Literature  in  the  University  0 

Pennsylvania,  and  an  Appendix,  showing  the  Pronunciation  of  nearly  3000  of 

the  oioat  important  Geographical  Names.     One  volume,  12mo. 

of  nearly  600  pages,  bound  in  Leather.     Price  #1 

Aaiong  tne  wants  of  our  time  was  a  good  dictionary  of  our  own  language,  especially  adapted 
for  academies  and  schools.  The  books  which  have  long  been  in  use  were  of  little  value  to  the 
junior  students,  being  too  concise  in  the  definitions,  and  unmethodical  in  the  arrangement 
Reid's  English  Dictionary  was  compiled  expressly  to  develop  the  precise  analogies  ana  various 
properties  of  the  authorized  words  in  general  use,  ty  the  standard  authors  apd  orators  who  uae 
our  vernacular  tongue. 

Exclusive  of  the  large  number  of  proper  names  which  are  appended,  this  Dictionary  includes 
four  especial  improvements — and  when  their  essential  value  to  the  student  is  considered,  the 
sterling  character  of  the  work  as  a  hand-book  of  our  language  will  be  instantly  perceived. 

The  primitive  word  is  distinguished  by  a  larger  type  ;  and  when  there  are  any  derivatives 
from  it,  they  follow  in  alphabetical  order,  and  the  part  of  speech  is  appended,  thus  furnishing  a 
complete  classification  of  all  the  connected  analogous  words  of  tlie  same  species. 

With  this  facility  to  comprehend  accurately  the  determinate  meaning  .,f  the  English  word,  in 
Conjoined  a  rich  illustration  for  the  linguist.  The  derivation  of  all  the  piimitive  words  is  dis- 
tinctly given,  and  the  phrases  of  the  languages  whence  they  are  deduced,  Whether  composite  or 
simple;  so  that  the  student  of  foreign  languages,  both  ancient  and  modern,  by  a  reference  to 
any  word,  can  ascertain  the  source  whence  it  has  been  adopted  into  our  own  form  of  speech. 
This  is  a  great  acquisition  to  the  person  who  is  anxious  to  use  words  in  their  utmost  clearness 
of  no  easing. 

To  these  advantages  is  subjoined  a  Vocabulary  of  the  Roots  of  English  Words,  which  is  of 
peculiar  value  to  the  collegian.  The  fifty  pages  which  it  includes,  furnish  the  linguist  with  a 
wide-spread  field  of  research,  equally  amusing  and  instructive.  There  is  also  added  an  Ac 
cented  List,  to  the  number  of  fifteen  thousand,  of  Greek,  Latin,  and  Scripture  Proper  Names. 

R  E  COMME  N  U  A  TI  ( )  N  S 

REID'S  Dictionary  of  the  English  Language  is  an  admirable  book  for  the  use  of  schools. 
Its  plans  combine  a  greater  number  ol  desirable  conditions  (or  euch  a  work,  than  any  \vhh 
which  I  am  acquainted:  and  it  seems  to  me  to  be  executed,  in  general  with  great  judgment, 
fidelity,  and  accuracy. 

C.  S.  HENRY, 
Professor  of  Philosophy,  History,  and  Belles  Lettrej, 

in  the  University  of  the  City  of  New-  York. 

Keif's  Dictionary  of  the  English  Language  is  compiled  upon  sound  principles,  and  with 
judgment  and  accuracy.  It  has  the  merit,  too,  of  combining  much  more  than  is  usually  lookad 
for  m  Dictionaries  of  small  size,  and  will,  I  believe,  be  found  excellent  as  a  convenient  manual, 
for  genera'  use  and  reference,  and  also  for  various  purposes  of  education. 

HENRY  REED, 

Professor  of  English  Literature  in  the   University  of  Pennsylvania 

After  a  careful  examination,  I  am  convinced  that  Reid's  English  Dictionary  has  stroni 
lainis  upon  the  attention  of  teachers  generally.  It  is  of  convenient  size,  beautifully  executed, 
nd  seems  well  adapted  to  the  use  of  scholars,  from  the  common  school  to  the  university. 

D.  H.  CHASE, 

Principal  of  Preparatory  School. 
MIDDLETOWX,  Ct. 

Af'er  a  thorough  examination  of"  Reid's  English  Dictionary,"  I  may  safely  say  that  I  con 
pider  it  superior  to  any  of  the  School  Dictionaries  with  which  I  am  acquainted.  Its  accurat* 
»«d  concise  definitions,  and  a  vocabulary  of  the  roots  of  English  words,  drawn  from  an  authoi 
sf  such  authority  as  Bosworth,  are  not  among  the  least  of  its  excellencies. 

M.  M.  PARKS, 
Professor  of  Ethics,  U.  S.  Military  Acadrmy,  West  Point 


Sibttra. 


GKEEK    OLLENDORFF; 

BEING   A   PROGEESSIVE   EXHIBITION    OF   THE   PEINCIPLSH 
OF   THE    GEEEK    GRAMMAR. 

Designed  for  Beginners  in  Greek,  and  as  a  Book  of  Exercises  for 
Academies  and  Colleges. 

BY   ASAHEL  C-   KENDRICK, 

Professor  of  the  Greek  Language  and  Literature  in  the  University  of  Rochester. 

One  volume,  12mo.    SI. 

Extract  from  the  Prefab. 

^  .ie  present  work  is  what  its  title  indicates,  strictly  an  Ollendotf,  and  aims  to  apply  the 
methods  which  have  proved  so  successful  in  the  acquisition  of  the  Modern  languages  to  the 
study  of  Ancient  Greek,  with  such  differences  of  course  as  the  different  genius  of  the  Greek, 
and  the  different  purposes  for  which  it  is  studied,  would  suggest.  It  differs  from  the  modern 
Otlendorffs  in  containing  Exercises  for  reciprocal  translation,  in  confining  them  within  a  smaller 
compass,  and  m  a  more  methodical  exposition  of  the  principles  of  the  language. 

It  differs,  on  the  other  hand,  from  other  excellent  elementary  works  in  Greek,  which  have 
recently  appeared,  in  a  more  rigid  adherence  to  the  Oilendorff  method,  and  the  greater  sim- 
plicity of  its  plan  :  in  simplifying  as  much  as  possible  the  character  of  the  Exercises,  and  in 
keeping  out  of  sight  every  thing  which  would  divert  the  student's  attention  from  the  naked  con- 
struction. 

The  object  of  the  Author  in  this  work  was  twofold ;  first,  to  furnish  a  book  which  should 
serve  as  an  introduction  to  the  study  of  Greek,  and  precede  the  use  of  any  Grammar.  It  will 
therefore  be  found,  although  not  claiming  to  embrace  all  the  principles  of  the  Grammar,  yet 
complete  in  itself,  and  will  lead  the  pupil,  by  insensible  gradations,  from  the  simpler  con- 
structions to  those  which  are  more  complicated  and  difficult. 

Tne  exceptions,  and  the  more  idiomatic  forms,  it  studiously  leaves  one  side,  and  only  aims 
to  exhibit  the  regular  and  ordinary  usages  of  the  language,  as  the  proper  starting  point  for  the 
student's  further  researches. 

In  presenting  these,  the  Author  has  aimed  to  combine  the  strictest  accuracy  with  the  utmost 
simplicity  of  statement.  He  hopes,  therefore,  that  his  work  will  find  its  way  among  a  younger 
class  of  pupils  than  have  usually  engaged  in  the  study  of  Greek,  and  will  win  to  the  acquisi- 
tion of  that  noble  tongue  many  in  our  Academies  and  Primary  Schools,  who  have  been  repelled 
by  the  less  simple  character  of  our  ordinary  text-books.  On  this  point  he  would  speak  ear- 
nestly. This  book,  while  he  trusts  if.  will  bear  the  criticism  of  the  scholar,  and  be  found 
adapted  to  older  pupils,  has  been  yet  constructed  with  a  constant  reference  to  the  wants  of  the 
young  ;  and  he  knows  no  reason  why  boys  and  girls  of  twelve,  ten,  or  even  eight  years  of  ase 
may  not  advantageously  be  put  to  the  study  of  this  book,  and,  under  skilful  instruction,  rapidly 
master  its  contents. 


GESENIUS'S  HEBREW  GKAMMAE. 

[Fourteenth  Edition,  as  revised  by  Dr.  E.  RODIGER.  Translated  by  T.  J.  CONANI 
Professor  of  Hebrew  iu  Madison  University,  N.  Y. 

With  the  Modifications  of  the  Editions  subsequent  to  the  Eleventh,  by  Dr.  DA  VIES 

of  Siepney  College,  London. 

To  which  are  added,  A  COTTRSE  OP  EXERCISES  IN  HEBREW  GRAMMAR,  and  a  HEBREW  CKRBA 

TOMATHY,  prepared  by  the  Translator.     One  handsomely  printed  vol.  8vo.     Price  $2. 

Extract  from  the.  Translator's  Preface. 

"The  fourteenth  editioff  of  the  Hebrew  Grammar  of  Gesenius  is  now  offered  to  the  public 
,   the  translator  t>l    he  eleventh  edition,  by  whom  this  work  was  first  made  accessible  to  stv 
ents  in  the  English  > inguage.     The  convicuon  expressed  in  his  pit-face  to  that  edition,  that  iu 
publication  ia  tins  -  nuiitry  would  subserve  the  interests  of  Hebrew  literature,  has  been  fully 
[sustained  by  the  result.     After  a  full  trial  of  the  merits  of  this  work,  both  in  America  and  i» 
IK'iglor.d,  its  republicanon  is  iv»w  dcin:iini("l  in  it«  lute  -i  ;m  1  MIO-:I  in:i':-->vnd  form." 


anil  1'nim. 


country  in  wmcn  me  great  uoinan  conqueror  conuuctcu  me  c 
scribes.  The  volume,  as  a  whole,  htwevcr,  appears  to  be  admi 
which  it  was  designed.  Its  .style  of  editing  ami  its  typographica 
Lincoln's  excellent  edition  of  Livy — a  work  which  -some  month 


C.  JULIUS   CJESAITS  COMMENTARIES 


GALLIC  WAR. 

With  English  Notes,  Critical  and  Explanatory ;  A  Ltxicori.  ^eograpnical  and 
Historical  Indexes,  &c. 

BY  REV.  J.  A.  SPENCER,  A.  M., 

Editor  of"  Arnold's  Series  of  Greek  and  La^in  Books,"  en. 

One  handsome  vol.  12mo,  with  Map.    Price  $1. 

Ti  e  press  of  Messrs.  Appleton  is  becoming  prolific  of  superior  editions  of  tha  classics  uwd 
in  schools,  and  the  volume  now  before  us  we  are  disposed  to  regard  as  one  of  the  nost  beautiful 
and  high'y  finished  among  them  all,  both  in  its  editing  and  its  execution.  The  classic  Latin  in  which 
the  greatest  general  and  the  greatest  writer  of  his  age  recorded  his  achievements,  has  leen  sadij 
corrupted  in  the  lapse  of  centuries,  and  its  restoration  to  a  pure  and  perfect  text  is  a  work  re- 
quiring nice  discrimination  and  sound  learning.  The  text  which  Mr.  Spencer  has  adopted  is  thai 
of  Oudendorp,  with  such  variations  as  were  suggested  by  a  careful  collation  of  the  leading  critict 
of  Germany.  The  notes  are  as  they  should  be,  designed  to  aid  the  labors  of  the  student,  not  to 
supei-sede  them.  In  addition  to  these,  the  volume  contains  a  sketch  of  the  life  of  Caesar,  a  briel 
Lexicon  of  Latin  words,  a  Historic^!  and  a  Geographical  Index,  together  with  a  map  of  the 
country  in_which  the  great  Roman  conqueror  conducted  the  campaigns  he  ?o  graphically  do- 

"mirably  suited  to  the  purpose  for 
ical  execution  reminds  us  of  Prof. 
inths  since  had  already  passed  to  a 

second  impression,  and  has  now  been  adopted  in  most  of  the  leading  schools  and  colleges  of  th« 
count ry. — Providence  JoiirnaJ. 

"  The  type  is  clear  and  beautiful,  and  the  Latin  text,  as  far  as  we  have  examined  it,  extremity 
accurate,  and  worthy  of  the  work  of  the  great  Roman  commander  and  historian.  No  one  edition 
lias  been  entirely  followed  by  Mr.  Spencer,  lie  has  drawn  from  Oudendorp,  Achaintre.  Lanuiiiv. 
Obcrliu,  Schneider,  and  Giani.  His  notes  are  drawn  somewhat  from  the  above,  and  al;--o  from 
Vossius.  navies,  Clarke,  and  S'.utgart.  These,  together  with  his  own  corrections  and  notes,  and 
an  excellent  lexicon  attache:!,  render  this  volume  the  most  complete  and  valuable  edition  ol 
Caesar's  Commentaries  yet  published. — Albany  Spectator. 

EXERCISES  IN  GREEK  PROSE  COMPOSITION. 

ADAPTED    TO    THE 

FIRST  BOOK  OF  XENOPHON'S  ANABASIS. 

BY  JAMES  R.  BOISE, 

Professor  in  Brown  University. 

One  volume,  12mo.    Price  seventy-five  cents. 

V  For  the  convenience  of  the  learner,  an  English-Greek  Vocabulary,  a  Catalogue  of  the  Im 
guiar  Verbs,  and  an  Index  to  the  principal  Grammatical  Notes  nave  been  appended. 

"A  school-book  of  the  highest  order,  containing  a  carefully  arranged  series  of  exercis«i  de 
rived  from  the  first  book  of  Xenophon's  Anabasis,  (which  is  appended  entire.)  an  Eng'ish  u4 
Greek  vocabulary  and  a  list  of  the  principal  modifications  of  irregular  verbs.  We  regard  it  M 
one  peculiar  excellence  of  this  book,  that  it  presupposes  both  the  diligent  scholar  and  the  paint 
taking  teacher,  in  ether  hands  it  would  be  not  only  useless,  but  unusable.  We  like  it  also,  b« 
enuse,  instead  of  aiming  to  give  the  pupil  practice  in  a  variety  of  styles,  it  places  before  him  bnl 
a  single  model  cf  Greek  composition,  and  that  the  very  author  who  combines  in  the  greatest  d> 
gree,  purity  of  language  and  idiom,  with  a  simplicity  that  both  invites  and  rewards  imitation.'' 
—  Christian  Register. 

•'  Mr.  Boise  is  Professor  of  Greek  in  Brown  University,  and  lias  prepared  these  exercisei 
as  an  accompaniment  to  the  First  Book  of  the  Anabasis  of  Xenophon  We  have  examined  tb 
plan  with  some  attention,  and  are  struck  with  its  utility.  The  exercises  consist  of  short  -* 


tences,  composed  of  the  words  used  in  the  text  of  the  Anabasis,  and  involving  the  same  construe 
rions;  and  the  system,  if  faithfully  pursued,  must  not  only  lead  to  familiarity  with  the  aulM) 
and  a  natural  adoption  of  his  style,  but  a!«>  to  sroat  ease  and  fanhless  excellence  in  Greek  our 
•  ~>£i'iti*i*i  '• — f>tntc#tfi.nt  Clnirr.hmun. 


A  MANUAL 


GRECIAN  AND  ROMAN  ANTiaUITIES. 

BY   DR.   E.    F.   BOJESEN, 

Professor  of  the  Greek  Language  and  Literature  in  the  University  of  Sora 
Translated  from  the  German. 

EDITED,    WITH    NOTES    AND   A    COMPLETE    SERIES  OF  QUESTIONS,  BY   TH« 

REV.  THOMAS  K.  ARNOLD,  M.  A. 
REVISED  WITH  ADDITIONS  AND  CORRECTIONS. 

One  neat  volume,  12mo.    Price  $1. 

The  present  Manual  of  Greek  and  Roman  Antiquities  is  far  superior  to  any  thing  on  flic 
*ame  topics  as  yet  offered  to  the  American  public.  A  principal  Review  of  Germany  says  : — 
Small  -AH  he  compass  of  it  is,  we  may  confidently  affirm  that  it  is  a  great  improvement  on  all 
preceding  worws  of  the  kind.  We  no  longer  meet  with  the  wretched  old  method,  in  which  sub- 
jects essentially  distinct  are  herded  toerether.  and  connected  subjects  disconnected,  but  hav«  t 
simple,  systematic  arrangement,  by  which  the  reaaer  easily  receives  a  clear  representaf'«in  i>( 
Roman  life.  We  ^»  longer  stumble  against  countless  errors  in  detail,  which  though  long  age 
assailed  and  extirpated  by  Niebuhrand  others,  have  found  their  last  place  of  refuge  in  nur  Ma- 
nuals. The  recent  investigations  of  philologists  and  jurists  have  been  extensively,  but  carefullj 
and  circumspectly  used.  The  conciseness  and  precision  which  the  author  has  every  where 
prescribed  to  himself,  prevents  the  superficial  observer  from  perceiving  the  essential  superiority 
of  the  book  to  its  predecessors,  but  whoever  subjects  it  to  a  careful  examination  will  discover 
(his  on  every  page." 

The  Editor  says : — "  I  fully  believe  that  the  pupil  will  receive  from  these  little  works  a 
correct  and  tolerably  complete  picture  of  Grecian  and  Roman  life;  what  I  may  call  the  POLI- 
TICAL portions — the  account  of  the  national  constitutions  and  their  effects — appear  to  me  to  be 
of  great  value;  and  the  very  moderate  extent  of  each  volume  admits  of  its  being  thoroughly 
mastered— of  it?  being  GOT  UP  and  RETAINED." 

"  A  work  long  needed  in  our  schools  and  colleges.  The  manuals  of  Rennet,  Adam,  Potter 
and  Robinson,  with  .je  more  recent  and  valuable  translation  of  Eschenburg,  were  entirely  too 
roluminous.  Here  is  ne  iher  too  much,  nor  too  little.  The  arrangement  is  admirable — every 
subject  is  treated  of  in  its  proper  place.  We  have  the  general  Geography,  a  succinct  historic^ 
vifcw  of  the  general  subject ;  the  chirography,  history,  laws,  manners,  customs,  and  religion  of 
each  State,  as  well  i'^the  points  of  union  for  all,  beautifully  arransed.  We  regard  the  work  aa 
the  ?ery  best  adjun?  to  classical  study  for  youth  that  we  have  seen,  and  sincerely  hope  tint 
wachers  may  be  bri  ^ht  to  regard  it  in  the  same  light.  The  whole  is  copiously  digested  int» 
ppropriate  questions." — S.  Lit.  Gazette. 

From  Professor  Lincoln,  of  Brown  University, 

"  I  found  on  my  table  after  a  short  absence  from  home,  your  edition  of  Bojecen's  Greek  an 
Roman  Antiquities.  Pray  accept  my  acknowledgments  for  it.  I  am  agreeably  surprised  to 
fLid  on  examining  it,  that  within  so  very  narrow  a  compass  for  so  comprehensive  a  subject,  the 
!>ook  contai  is  so  much  valuable  matter;  and,  indeed,  so  far  as  I  see,  omits  noticing  no  topics  es- 
tential.  It  will  be  a  very  useful  book  in  Schools  and  Colleges,  and  it  is  far  superior  to  any  thing 
that  I  know  of  the  same  kind.  Besides  being  cheap  and  accessible  to  all  students,  it  lias  the 
fttat  msrit  of  discussing  its  topics  in  a  consecutive  and  connected  manner." 

Extract  of  a  letter  from  Proftssor  Tyler,  of  Amherst  College. 

"  I  have  never  found  time  till  lately  to  look  over  Bojeson's  Antiquities,  of  which  you  were 
tind  enough  to  send  me  a  copy.  I  think  it  an  excellent  book ;  learned,  accurate,  concise,  and 
perspicuous;  well  adapted  for  use  iiv  'he  Academy  or  the  College,  and  comprehending  in  » 
•>Tial!  compass,  more  «^n  in  valuable  on  the  subject  than  many  extended  treatises  " 

3 


frat 
CICERO  DE    OFFICIIS. 

WITH  ENGLISH  NOTES. 

Chiefly  selected  and  translated  from  the  editions  of  Zumpt  and  Bonneli. 

BY  THOMAS  A.  THACHER, 

Assistant  Professor  of  Latin  in  Yale  College. 
One  volume  12mo.     90  cents. 

This  edition  of  De  OPkiin  has  the  advantage  over  any  other  with  which  we  are  acquainted, 
ai'mcre  copious  notes,  txjuer  arrangement,  and  a  more  beautiful  typography.  The  text  ci 
Zumpt  appears  to  have  be'.-n  c'osely  followed,  except  in  a  very  few  instances,  where  it  is  varied 
on  the  authority  of  Beier,  O/elli  and  Bonneli.  Teachers  and  students  will  do  wtll  to  examine 
this  edition. 

"Mr.  Thacher  very  modestly  disclaims  for  himielf  more  than  the  cirlit  cf  a  compiler  and 
translator  in  me  editing  of  this  work.  Being  ourselves  unblessed  with  vhe  works  of  Zumpt, 
Bonneli,  and  other  Gorman  writers  to  whom  Mr.  T.  credits  most  of  his  notes  and  comments,  we 
cannot  affirm  that  more  credit  is  due  him  than  he  claims  for  his  tabors,  but  we  may  accord  him 
the  merit  of  an  extte'nely  judicious  and  careful  compiler,  if  no  more ;  'for  we  have  seen  no  re- 
mark without  an  important  bearing,  nor  any  point  requiring  elucidation  which  was  passed  un- 
noticed. 

"  This  work  of  Cicero  cannot  but  interest  eveiry  one  at  all  disposed  to  inquire  into  the  views 
of  the  ancients  on  morals. 

"This  valuable  philosophical  treatise,  emanating  from  the  pen  of  the  illustrious  Roman,  de- 
rives a  peculiar  interest  from  the  fact  of  its  being  written  with  the  object  to  instruct  his  son,  of 
whom  me  author  had  heard  unfavorable  accounts,  and  whom  the  weight  ot  his  public  duties 
had  prevented  him  from  visiting  in  person.  It  presents  a  great  many  wise  maxims,  apt  and 
rich  illustrations,  and  the  results  of  the  experience  and  reflections  of  an  acute  and  powerful 
mind.  It  is  well  adapted  to  the  use  of  the  student  by  copious  and  elaborate  notes,  explanatory 
of  the  text,  affording  ample  facilities  to  its  entire  comprehension.  These  have  been  gleaned 
•fith  great  ludgment  from  the  most  learned  and  reliable  authorities, — such  as  Zumpt,  Bonneli, 
and  others".  Mr.  Thacher  has  evinced  a  praiseworthy  care  and  diligence  in  preparing  the  vo- 
lume for  the  purposes  for  which  it  was  designed." 


SELECT  ORATIONS  OF  M.  TULLIUS  CICERO  • 

WITH  NOTES,  FOR  THE  USE  OF  SCHOOLS  AND  COLLEGES. 

BY  E.  A.  JOHNSON, 

Professor  of  Latin  in  the  University  of  New-  York. 

One  volume,  12mo.    $1. 

"  This  edition  of  Cicero's  Select  Orations  possesses  some  special  advantages  for  the  student 
which  are  both  new  and  important.  It  is  the  only  edition  which  contains  the  improved  text 
that  has  been  prepared  by  a  recent  careful  collation  and  correct  deciphering  of  the  best  manu 
scripts  of  CICERO'B  writings.  It  is  the  work  of  the  celebrated  ORELLI,  together  with  that  of 
MADVIO  and  KT,OTZ,  and  has  been  done  since  the  appearance  of  ORELLI'S  complete  edition.  The 
Notes,  by  Proftesor  JOHNSON,  of  the  New-York  University,  have  been  chiefly  selected,  with  great 
care,  from  the  best  German  authors,  as  well  as  the  English  edition  of  ARNOLD.  Although 
abundant,  and  almost  profuse,  they  yet  appear  generally  to  relate  to  some  important  point  in 
the  text  or  subject,  which  the  immature  mind  of  pupils  could  not  readily  detect  without  aid. 
We  do  not  know  how  a  more  perfect  edition  for  the  use  of  schools  could  well  be  prepared." 

"This  is  a  beautiful  and  most  excellent  edition  of  the  great  Roman  orator;  and,  so  far  as 
we  know,  the  best  ever  published  in  this  country.  It  contains  the  four  orations  against  Cata- 
/ine,  the  oration  for  the  Monilian  Law,  the  oration  for  Marcellus,  for  Ligarius,  for  King  Deio- 
Urius,  for  the  poet  Archias,  and  for  Milo.  In  preparing  the  text  of  these  orations  the  editor  has 
a  .'ailed  himself  of  the  best  German  and  English  editions  ;  and  the  notes  have  been  gathered 
from  every  available  source.  These  are  so  abundant  —  filling  more  than  300  pages  —  as  to  leave 
almost  nothing  to  be  desired  by  the  student.  They  are  philological,  explanatory  and  historical. 
Each  Oration  la  furnished  with  a  valuable  Introduction,  containing  what  is  necessary  for  the 


student  to  know  preparatory  to  the  commencement  of  the  study  of  the  Oration,  and  an  analysis 
of  the  plan  and  argument  of  each  Oration.      Furnished   with  this  edition  of  Cicero's  Select 


. 

Orations,  the  student  is  orepared  to  enter  with  pleasure  and  profit  on  the  study  of  this  eleganl 
«uid  renownefl  classic  author.1'  —  Uosivrt  Atlas. 


MANUAL 

OF 

ANCIENT  GEOGRAPHY  AND  HISTORY. 


BY  WILHELM   PUTZ, 

PRINCIPAL   TUTOR   IN   THE    GYMNASIUM    OF    PUREN 

Translated  from  the  German. 
EDITED    BY  TliE    REV.    THOMAS   K.    ARNOLD,   >1  A., 

AUTHOR   OP   A    SERIES    OP   "GREEK   AND    LATIN   TEXT-BOOKS." 

One  volume,  12mo.    $1. 


'•  At  no  perioc  nag  History  presented  such  strong  claims  upon  the  attention  of  the  learned,  a» 
it  the  present  day  ;  and  to  no  people  were  its  lessons  of  such  value  as  to  those  of  the  United 
Suites.  With  no  past  of  our  own  to  revert  to,  the  great  masses  of  our  better  educated  are  tempted 
'.o  overlook  a  science,  which  comprehends  all  others  in  its  grasp.  To  prepare  a  text-book,  which 
shall  present  a  full,  clear,  and  accurate  view  of  the  ancient  world,  its  geography,  its  political, 
civil,  social,  religious  state,  must  be  the  result  only  of  vast  industry  and  learning.  Our  exami- 
nation of  the  present  volume  leads  us  to  believe,  that  as  a  text-book  on  Ancient  History,  for  Col- 
leges and  Academies,  it  is  the  best  compend  yet  published.  It  bears  marks  in  its  methodical 
arrangement,  and  condensation  of  materials,  01  the  untiring  patience  of  German  scholarship ;  and 
m  its  progress  through  the  English  and  American  press,  has  been  adapted  for  acceptable  use  in 
our  best  institutions.  A  noticeable  feature  of  the  book,  is  its  pretty  complete  list  of  'sources  ol 
information'  upon  the  nations  which  it  describes.  This  will  be  an  invaluable  aid  to  the  student 
in  his  future  c«urse  of  reading." 

"  Wilhelm  Piitz;  the  author  of  this  '  Manual  of  Ancient  Geography  and  History,'  is  Principa1 
Tutor  (  Oberleher)  in  the  Gymnasium  of  Duren,  Germany.  Hfs  book  exhibits  the  advantages  o 
tha  German  method  of  treating  History,  in  its  arrangement,  its  classification,  and  its  rigid  analy- 
sis. The  Manual  is  what  it  purports  to  be, '  a  clear  and  definite  outline  of  the  history  of  the 
principal  nations  of  antiquity,'  into  which  is  incorporated  a  concise  geography  of  each  country. 
The  work  is  a  text- book ;  to  be  studied,  and  not  merely  read  It  is  to  form  the  groundwork  oj 
subsequent  historical  investigation, — the  materials  of  which  are  pointed  out,  at  the  proper  places, 
in  the  Manual,  in  careful  references  to  the  works  which  treat  of  the  subject  directly  under  con- 
sideratnn.  The  list  of  references  (especially  as  regards  earlier  works)  is  quite  complete,— thus 
supplying  that  desideratum  in  Ancient  History  and  Geography,  which  has  been  supplied  so  fully 
Ly  1)  r.  J.  C.  I.  Gieseler  in  Ecclesiastical  History. 

11  The  nations  whose  history  is  considered  in  the  Manual,  are :  in  Asia^  the  Israelites,  th?  In- 
dians, the  Babylonians,  the  Assyrians,  the  Medes,  the  Persians,  the  Phoenicians,  the  States  of  Asia 
Minor ;  in  Africa,  the  Ethiopian,  the  Egyptians,  the  Carthaginians ;  in  Europe,  the  Greeks,  the 
Macedonians,  the  Kingdoms  which  arose  out  of  the  Macedonian  Monarchy,  the  Romans.  The 
ord  ?r  in  which  the  history  of  each  is  treated,  is  admirable.  To  the  whole  are  appended  a  '  Chro 
r.ological  Table,'  and  a  well-prepared  series  of  'Questions.'  The  pronunciation  of  prope* 
Games  is  indicated,— an  excellent  feature.  The  accents  are  given  with  remarkable  correctness. 
Ylitt  typographical  execution  of  the  American  edition  is  most  excellent."— S.  W.BaptistChronicle 

'•  lake  every  thing  which  proceeds  from  the  editorship  of  that  eminent  Instructor,  T.  K.  Arnold, 
this  Manual  appears  to  be  well  suited  to  the  design  with  which  it  was  prepared,  and  will,  un- 
doubtedly, secure  for  itself  a  place  among  the  text-books  of  schools  and  academies  thoughout  th« 
of-untry.  It  presents  an  outline  of  the  history  of  the  ancient  nations,  from  the  earliest  ages  to  tha 
fall  of  the  Western  Empire  in  the  sixth  century,  the  events  being  arranged  in  the  order  of  an 


??h  jr. 

"  ft  was  originally  prepared  by  Wilhelm  Piitz,  an  eminent  German  scholar,  and  translated  and 
edited  in  England  by  Rev.  T.  K.  Arnold,  and  is  now  revised  and  introduced  to  the  American 
public  in  a  well  written  preface,  by  Mi  Georjro  W.  Greene,  Teacher  of  Modern  Language*  ii 
Brown  University  "— Prov.  Journal. 


HAND  BOOK 


OF 

MEDIAEVAL   GEOGRAPHY  AND   HISTORY 

BY 

WILHELM    PUTZ, 

PRINCIPAL  TUTOR  IN  THE  GYMNASIUM  OF  DUREN 

Translated  from  the  German  by 
REV,  R,  B,  PAUL,  M,  A,, 

Vicar  of  St.  Augustine's,  Bristol,  and  late  Fellow  of  Exeter  Colle&t,  Oxford, 
1  volume,  12mo.    75  cts. 

HEADS    OF    CONTENTS. 

I.  Germany  before  the  Migrations. 
II.  The  Migrations. 

TUP    MIDDLE    AGES. 

FIRST  PERIOD.—  From  the  Dissolution  of  the  Western  Empire  to  the  Accession 
gitins  and  Abbasides. 

SECOND  PERIOD. — From  the  Accession  of  the  Carlovingians  and  Abbasides  to  the  first  CrusacK. 

THIRD  PERIOD.— Age  of  the  Crusades. 

FOURTH  PERIOD. — From  the  Termination  of  the  Crusades  to  the  Discovery  of  America. 

"  The  characteristics  of  this  volume  are :  precision,  condensation,  and  luminous  arrangement 
It  is  precisely  what  it  pretends  to  be — a  manual,  a  sure  and  conscientious  guide  for  the  studeni 
through  the  crooks  and  tangles  of  Mediaeval  history.  *  *  *  *  All  the  great  principles  of  th^i 
ex'ensi'a  Period  are  carefully  laid  down,  and  the  most  important  facts  skilfully  grouped  aiouml 
them.  There  is  no  period  of  History  for  which  it  is  more  difficult  to  prepare  a  work  like  this 
and  none  for  which  it  is  so  much  needed.  The  leading  facts  are  well  established,  but  they  are 
scattered  over  an  immense  space ;  the  principles  are  ascertained,  but  their  development  wa? 
slow,  unequal,  and  interrupted.  There  is  a  general  breaking  up  of  a  great  body,  and  a  paiceiling 
of  it  out  among  small  tribes,  concerning  whom  we  have  only  a  few  general  data,  and  are  left  tr 
analogy  and  conjecture  for  the  details.  "Then  come  successive  attempts  at  organization,  each 
more  or  less  independent,  and  all  very  imperfect.  At  last,  modern  Europe  begins  slowly  m 
emerge  from  the  chaos,  bat  still  under  forms  which  the  most  diligent  historian  cannot  always 
comprehend.  To  reducs  such  materials  to  a  clear  and  definite  form  is  a  task  of  no  small  dinY 
culty,  and  in  which  partial  success  deserves  great  praise.  It  is  not  too  much  to  say  that  't  has 
never  been  so  well  done  within  a  compass  so  easily  mastered,  as  in  the  little  volume  wh»^h  is> 
now  offered  to  the  public." — Extract  Jrom  American  Preface. 

"This  translation  of  a  foreign  school-book  embraces  a  succinct  and  well  ar  anged  body  of 
facts  concerning  European  and  Asiatic  history  and  geography  during  the  middle  ages.  It  in 
furnished  with  printed  questions,  and  it  seems  to  b<3  well  adapted  to  its  purpose,  in  all  respect* 
The  rnedlajval  period  is  one  of  the  most  interesting  in  the  annals  of  the  world,  and  a  knowled^.' 
of  its  great  men,  and  of  its  progress  in  arts,  arms,  government  and  religion,  is  particularly  ini 
portan',  since  this  period  is  the  basis  of  our  own  social  polity." — Commercial  Advertiser. 

"  This  is  an  immense  amount  of  research  condensed  into  a  moderately  sized  volume,  in  a  wa> 
which  no  one  has  patience  to  do  but  a  German  scholar.  The  beauty  of  the  work  is  its  luminoi;. 
arrangjinent.  It  is  a  guide  to  the  student  amidst  the  intricacy  of  Mediaeval  History,  the  me? 
difficult  period  of  the  world  to  understand,  when  the  Roman  Empire  was  breaking  up  and  p/n 
celling  out  into  smaller  kingdoms,  and  every  thing  was  in  a  transition  state.  It  was  a  penou  > 
chaos  from  v;hich  modern  Europe  was  at  length  to  arise. 

The  author  has  briefly  taken  up  the  principal  political  and  social  influences  whicr-  wert 
acting  on  society,  and  shown  their  bearing  from  the  tim?,  previous  to  the  migrations  o!  the 
Northern  nations,  down  through  the  middle  ages  to  the  sixteenth  century.  The  n  >tes  on  the 
crusader  are  particularly  valuable,  and  the  range  of  observation  embraces  not  only  Euicpi  but 
the  East.  To  the1,  student  it  will  be  a  most  valuable  Hand-book,  savin?  him  a  world  of  trouble 
to  huntias?  up  nii'horiiic^  and  facts."  Rrv  Dr.  Kip.  in  Albam/  State  Refisler. 

4 


fnglis] 
A  MANUAL  Of  ANCIENT  AND  MODERN  HISTORY, 

COMPRISING: 

I.  ANCIENT  HISTORT,  containing  the  Political  History,  Geographical"  Position,  and  Soci» 
State  of  the  Principal  Nations  of  Antiquity,  carefully  aigested  from  the  Ancient  Writers,  and  il 
!ustra'ed  by  the  discoveries  of  Modern  Travellers  and  Scholars. 

II.  MODERN  His  TORY,  containing  the  Rise  and  Progress  of  the  principal  European  Nation*, 
their  Political  History,  and  the  changes  in  their  Social  Condition:  with  a  History  of  the  Colonial 
F^vcded  by  Europeans.  By  W.  COOKE  TAYLOR,  LL.D.,  of  Trinity  College,  Dublin.    Revised, 
Ki&  Additions  on  American  History,  by  C.  S.  Henry,  D.  D.,  Professor  of  History  in  the  Univer 
city  of  N.  Y.,  and  Questions  adapted  for  the  Use  of  Schools  and  Colleges.    One  handsome  rol., 
§v  i,  ol  800  pages,  $'.],25 ;  Ancient  History  in  1  vol.  $1,25,  idodern  History  in  1  vol.,  $1,50. 

The  ANCIENT  IIISTORY  division  comprises  Eighteen  Chapters,  whu  :i  include  the  general 
outlines  of  the  History  of  Egypt — the  Ethiopians — Babylonia  and  Assyria — Western  Asia — Pal- 
estine— tlie  Empire  of  the  Medes  and  Persians — Phoenician  Colonies  in  Northern  Africa — Founts 
ation  and  History  of  the  Grecian  States— Greece— the  Macedonian  Kingdom  and  Empire— the 
Biatea  that  arose  irom  the  dismemberment  of  the  Macedonian  Kingdom  and  Empire — Ancient 
Italy — Sicily — the  Roman  Republic — Geographical  and  Political  Condition  of  the  Roman  Emoirt 
—History  of  the  Roman  Empire— and  India— with  an  Appendix  of  important  illustrative  articles 

This  portion  is  one  of  the  best  Compends  of  Ancient  History  that  ever  yei  has  appeared  li 
contains  a  complete  text  for  the  collegiate  lecturer ;  and  is  an  essential  hand-book  for  the  student 
who  is  desirous  to  become  acquainted  with  all  that  is  memorable  in  general  secular  archaeology. 

The  MODERN  HISTORY  portion  is  divided  into  Fourteen  Chapters,  on  the  following  genera] 
subject*:— Consequences  of  the  Fall  of  the  Western  Empire— Rise  and  Establishment  of  the 
Saracenic  Power — Restoration  of  the  Western  Empire — Growth  of  the  Papal  Power — Revival 
of  Literature — Progress  of  Civilization  and  Invention — Reformatioa,  and  Commencement  of  tht 
States  System  in  Europe— Augustan  Ages  of  England  and  France— Mercantile  and  Colonial  Sys- 
tem— Age  of  Revolutions — French  Empire — History  of  the  Peace — Colonization — China — the 
Jews— with  Chronological  and  Historical  Tables  and  other  Indexes.  Dr.  Henry  has  appended  a 
new  chapter  on  the  History  of  the  United  States. 

This  Manual  of  Modern  History,  by  Mr.  Taylor,  is  the  most  valuable  and  instructive  work 
concerning  the  general  subjects  which  rt  comprehends,  that  can  be  found  in  the  whole  department 
of  historical  literature.  Mi.  Taylor's  book  is  fast  superseding  all  other  compends,  and  is  already 
adopted  as  a  text-book  in  Harvard,  Columbia,  Yale,  New-York,  Pennsylvania  and  Brown  Uiu- 
veraities,  and  several  leading  Academies. 


LECTURES 

ON 

MODERN  HISTORY. 

By  THOMAS  ARNOLD,  D.D., 

Regius  Professor  of  Modern  History  in  the  University  of  Oxford^  and  H&uL 
Master  of  Rugby  School. 

EDITED,    WITH   A    PREFACE    AND    NOTES, 

By  HENRY  REED,  LL.D., 
Profcttor  of  English  Literature  in  the  University  tf  Pu. 

One  volume,  12mo.    $1,26. 

Extract  from  the  American  Editor's  Preface. 

m  preparing  this  edition,  I  have  had  in  view  its  use,  not  only  for  the  general  reader,  but  a  • 
Bb  <<  text-book  in  education,  especially  in  our  college  course  of  study.  *  '     *  The  introduction  of 
<k*  work  ai"  a  text-book  I  regard  as  important,  because,  as  far  as  my  information  entitles  me  to 
,  theie  is  no  book  better  calculated  to  inspire  an  interest  in  historical  study.    That  H  nai 


this  power  over  the  minds  of  students  I  can  say  from  experience,  which  enables  me  also  to  add. 
that  I  have  found  it  excellentiv  suited  to  a  course  of  college  instruction.  By  intelligent  and  en- 
"uprising  members  of  a  class  especially,  it  is  studied  as  a  text-book  with  zeal  and  animation. 

11 


MANUAL 


MODERN  GEOGRAPHY  AND  HISTORY. 

BY  W1LHELM  PUTZ, 

Author  of  Manuals  of  "  Ancient  Geography  and  History,"   "  Mediaval  Geogi  aphy  ana 

History,"  $c. 

TRANSLATED  FROM  THE  GERMAN.   REVISED  AND  CORREOTFD. 

One  volume,  12mo.      $>1. 

"Preface.— The  present  volume  completes  the  series  of  Professor  Piitz's  Handbooks  o\ 
Ancient,  Mediaeval,  and  Modern  Geography  and  History.  Its  adapta'ion  to  the  wants  of  tha 
student  will  be  found  to  be  no  less  complete  than  was  to  be  expected  from  the  fcrmer  Parta, 
which  have  been  highly  approved  by  the  public,  and  have  been  translated  into  several  lai> 
guages  besides  the  English.  The  difficulty  of  compressing  within  the  limits  of  a  single  volume 
the  vast  amount  of  historical  material  furnished  by  the  progress  of  modern  states  and  nations 
in  power,  wealth,  science,  and  literature,  will  be  evident  to  all  on  reflection  ;  and  they  wiK 
find  occasion  to  admire  the  skill  and  perspicacity  of  the  Author  of  this  Handbook,  not  only  in 
the  arrangement,  but  also  in  the  facts  and  statements  which  he  has  adopted. 

"  In  the  American  edition  several  improvements  have  been  made ;  the  sections  relating  to 
America  and  the  United  States  have  been  almost  entirely  re-written,  and  materially  enlarged 
and  improved,  as  seemed  on  every  account  necessary  and  proper  in  a  work  intended  for  general 
use  in  this  country ;  on  several  occasions  it  has  been  thought  advisable  to  make  certain  verbal 
corrections  and  emendations ;  the  facts  and  dates  have  been  verified,  and  a  number  of  explan- 
atory notes  have  been  introduced.  It  is  hoped  that  the  improvements  alluded  to  will  be  found 
to  add  to  the  value  of  the  present  Manual." 


FIEST  LESSONS  IN  COMPOSITION. 

IN   WHICH   THE    PRINCIPLES    OP   THE    ART    ARE    DEVELOPED   IN    CONNECTION    WITH 
THE    PRINCIPLES    OP    GRAMMAR; 

Embracing  full  Directions  on  the  subject  of  Punctuation:  with  copious 

Exercises. 

BY.  G.  P.  QUACKENBOS,  A.M. 

Rector  of  the  Henry  Street  Grammar  School,  N.  Y. 

One  volume,  12mo.    45  cts. 

EXTRACT   FROM  PREFACE. 

1  A  county  superintendent  of  common  schools,  speaking  of  the  important  branch  of  com 
position,  uses  the  following  language  :  'Fora  long  time  1  nave  noticed  with  regret  the  almost 
entire  neglect  of  the  art  of  original  composition  in  our  common  schools,  and  the  want  of  a 
proper  text  book  upon  this  essential  branch  of  education.  Hundreds  graduate  from  our  common 
schools  with  no  well-defined  ideas  of  the  construction  of  our  language.'  The  writei  imgut 
have  gone  further,  and  said  that  multitudes  graduate,  not  only  from  common  schools,  but 
from  some  of  our  best  private  institutions,  utterly  destitute  of  all  practical  acquaintance  with 
the  subject ;  that  to  many  such  the  composition  of  a  single  letter  is  an  irksome,  to  some  an 
almost  impossible  task.  Yet  the  reflecting  mind  must  admit  that  it  is  only  this  practical  appli- 
cation of  grammar  that  renders  that  art  useful— that  parsing  is  secondary  to  composing,  and 
the  analysis  of  our  language  almost  unimportant  when  compared  with  its  synthesis. 

"One  great  reason  of  the  neglect  noticed  above,  has,  no  doubt,  been  the  want  of  a  suitable 
text-book  on  the  subject.  During  the  years  of  the  Author's  experience  as  a  teacher,  he  ha 
examined,  and  practically  tested  the  various  works  on  composition  with  which  he  has  met. 
the  result  has  been  a  conviction  that,  while  there  are  several  publications  well  calculated  to 
advance  pupils  at  the  age  of  fifteen  or  sixteen,  there  is  not  one  suited  to  the  comprehension 
of  thos-'e  between  nine  and  twelve ;  al  which  time  it  is  his  decided  opinion  that  this  branch 
should  be  taken  up.  Heretofore,  the  teacher  has  been  obliged  either  to  make  the  scholar  labor 
through  a  work  entirely  too  difficult  for  him.  to  give  him  exercises  not  founded  on  any  regulai 
system,  or  to  abandon  the  branch  altogether— and  the  disadvantages  of  either  of  these  courses 
are  at  once  apparent. 

"  Tt  'f  'his  conviction,  founded  on  the  experience  not  only  of  the  Author,  but  of  many 
olher  teachers  with  whom  he  has  c  u^uitn  !,  ili.-jt  has  1H  f>  th*  production  of  toe  work  now 
afler^  I  to  ihe  pub'ic.  It  claims  to  be  a  first-book  in  composition,  and  is  intended  to  initiate 
•he  ••  Dinner.  t<y  eapv  and  pleasant  steps,  into  that  aii  important,  but  hitherto  <renei.illy  nc? 
eciet4.  art  " 


ENGLISH    SYNONYMES, 

CLASSIFIED  AND  EXPLAINED, 

WITH 

PRACTICAL   EXERCISES. 

DESIGNED    FOR    SCHOOLS    AND    PRIVATE   TUITION 
BY    G.    F.    GRAHAM, 

Author  of '  English,  or  the  Art  of  Composition,'  &c. 
WITH    AN    INTRODUCTION    AND    ILLUSTRATIVE    AUTHORITIES^ 

BY    HENRY    REED,    LL.D., 

Prof,  of  English  Literature  in  the  University  of  Penn. 

One  neat  Vol.   I2mo.  $1. 

CONTENTS.— SECTION  I.  Generic  and  Specific  Synonymes.  fl.  A.ctire 
and  Passive  Synonymes.  III.  Synonymes  of  Intensity.  IV.  Positive 
and  Negative  Synonymes.  V.  Miscellaneous  Synonymes.  Index  to 
Synonymes.  General  Index. 

Extract  from  American  Introduction. 

"  This  treatise  is  republished  and  edited  with  the  hope  that  it  will  be  found  useful  as  a  ten 
booK  in  the  study  of  our  own  language.  As  a  subject  of  instruction,  the  study  of  the  English 
tongue  does  not  receive  that  amount  of  systematic  attention  which  is  due  to  it,  whether  it  be 
combined  or  no  with  the  study  of  the  Greek  and  Latin.  In  the  usual  courses  of  education,  it  haa 
no  larger  scope  than  the  study  of  some  rhetorical  principles  and  practice,  and  of  grammatical 
rules,  which,  for  the  most  part,  are  not  adequate  to  the  composite  character  and  varied  idiom  of 
English  speech.  This  is  far  from  being  enough  to  give  the  needful  knowledge  of  what  is  the 
living  language,  both  of  our  English  literature  and  of  the  multiform  intercourse — oral  and  writ- 
ten — of  our  daily  lives.  The  language  deserves  better  care  and  more  sedulous  culture  |  it  needs 
much  more  to  preserve  its  purity,  and  to  guide  the  progress  of  its  life.  The  young,  instead  of 
havin?  only  such  familiarity  with  their  native  speech  as  practice  without  method  or  theory  gives, 
should  be  so  taught  and  trained  as  to  acquire  a  habit  of  using  words— whether  with  the  voice  or 
the  pen — fitly  and  truly,  intelligently  and  conscientiously." 

''For  such  training,  this  book,  it  is  believed,  will  prove  serviceable.  The  '•Practical  Exer- 
cisesy  a'tached  to  the  explanations  of  the  words,  are  conveniently  prepared  for  the  routine  of 
instruction.  The  value  of  a  course  of  this  kind,  resularly  and  carefully  completed,  will  be  more 
than  the  amount  of  information  gained  respecting'the  words  that  are  explained.  It  will  tend  to 
produce  a  thoughtful  and  accurate  use  of  language,  and  thus  may  be  acquired,  almost  uncon- 
sciously, that  which  is  not  only  a  critical  but  a  moral  habit  of  mind — the  habit  of  giving  utter- 
ance to  truth  in  simple,  clear  and  precise  terms — of  telling  one's  thoughts  and  leelings  in  words 
that  express  nothing  more  and  nothing  less.  It  is  thus  that  we  may  learn  how  to  escape  tlie 
evils  of  vagueness,  obscurity  and  perplexity— the  manifold  mischiefs  of  words  used  thought- 
lessly and  at  random,  or  words  used  in  ignorance  and  confusion. 

"In  preparing  this  edition,  it  seemed  to  me  that  the  value  and  literary  interest  of  the  book 
might  be  increased  by  the  introduction  of  a  series  of  illustrative  authorities.  It  is  in  the  addi- 
tion of  these  authorities,  containea  within  brackets  under  each  title,  and  also  of  a  general  index 
to  facilitate  reference,  that  this  edition  differs  from  the  original  edition,  which  in  other  respects 
is  exactly  reprinted.  I  have  confined  my  choice  of  authorities  to  poetical  quotations,  chiefly  be- 
cause i:  is  in  poetry  that  language  is  found  in  its  highest  purity  and  perfection.  The  selections 
have  b«-en  made  from  three  of  the  English  poets— each  a  great  authority,  and  each  belonging  to 
a  different  period,  so  that  in  this  way  some  historical  illustration  of  the  language  is  given  at 
ihe  same  time.  The  quotations  from  Shakspeare  (born  A.  D.  1564,  died  1616)  may  be  considered 
as  illustrating  the  ise  of  the  words  at  the  close  of  the  16th  and  besinning  of  the  17th  century; 
those  from  Milton  (born  1608,  died  1674)  the  succeeding  half  celitury,  or  middle  of  the  17th 
•>«nttry;  and  those  from  Wordsworth  (born  1770)  the  contemporary  use  in  the  19rh  eontury 

I  1 


foglislj. 
JHE    SHAKSPEARIAN    READER; 

A  COLLECTION  IF  THE  MOST  APPROVED  PLAYS  OF 

S  H  AKSPE  ARE. 

Cwi^-^lly  Revised,  wicn  Introductory  and  Explanatory  Notes, and  a  MemoJbt 

of  the  Author      Prepared  expressly  for  the  use  of  Classes, 

and  the  Family  Reading  Circle. 

BY  JOHN  W.  S.  HOWS, 
Professor  of  Elocution  in  Columbia  College. 

The  MAN,  whom  Nature's  self  hath  made 

To  mock  herself,  and  TRUTH  to  imitate.— Spenser. 

One  Volume,  12mo,  $1  25. 

At  a  oriod  when  the  fame  of  Shakspeare  is  "  striding  the  world  like  a  co.osaus,  '•  and  t«ti 
ticiwof  his  works  are  multiplied  with  a  profusion  thai  testifies  the  desire  awakened  in  all  classes 
jf  society  to  read  and  study  his  imperishable  compositions, — there  needs,  perhaps,  but  little 
apology  for  the  following  selection  of  his  works,  prepared  expressly  to  render  them  unezcep 
tionable  for  the  use  of  Schools,  and  acceptable  for  Family  reading.  Apart  from  the  fact,  that 
Shakspeare  is  the  "well-spring"  from  which  may  be  traced  the  origin  of  the  purest  poetry  in 
our  language, — a  long  course  of  professional  experience  has  satisfied  me  that  a  necessity  exiete 
for  the  addition  of  a  vr°rk  like  the  present,  to  our  stock  of  Educational  Literature.  Hits  writings 
are  peculiarly  adapted  for  the  purposes  of  Elocutionary  exercise,  when  the  system  of  instruction 
pursued  by  the  Teacher  is  based  upon  the  true  principle  of  the  art,  viz. — a  careful  analysis  of 
the  structure  and  meaning  of  language,  rather  than  a  servile  adherence  to  the  arbitrary  and  me- 
chanical rules  of  Elocution. 

To  impress  upon  the  mind  of  the  pupil  that  words  are  the  exposition  of  thought,  and  that  in 
reading,  or  speaking,  every  shade  of  thought  and  feeling  has  its  appropriate  shade  of  modulated 
tone,  ought  to  be  the  especial  aim  of  every  Teacher;  and  an  author  like  Shakspeare,  whose 
every  line  embodies  a  volume  of  meaning,  should  surely  form  one  of  our  Elocutionary  Text 
Books.  *  '  Still,  in  preparing  a  selection  of  his  works  for  the  express  purpose  contem- 
plated in  my  design,  I  have  not  hesitated  to  exercise  a  severe  revision  of  his  language,  beyond 
that  adopted  in  any  similar  undertaking — "  Bowdler's  Family  Shakspeare  "  not  even  excepted;  - 
and  simply,  because  I  practically  know  the  impossibility  of  introducing  Shakspeare  as  a  Cl»*» 
Book,  or  as  a  satisfactory  Reading  Book  for  Families  without  this  precautionary  reviai  ro.  • 
Extr-Mtfrom  the  Preface. 


HISTORY  MD  GEOGRAPHY 

OP 

THE     MIDDLE     AGES 

(CHIEFLY  FROM  THE  FRENCH.) 

BY  G.W.GREENE, 

Instructor  in  Brown   University. 

PART  I :  HISTORY.    One  volume,  12mo.    SI. 

Extract  from  Preface. 

"This  volume,  as  the  title  indicates,  is  chiefly  taken  from  a  popular  French  work,  which 
4U)  rapidly  passed  through  several  editions,  and  received  the  sanction  of  the  University.  It 
Will  be  found  to  contain  a  clear  and  satisfactory  exposition  of  the  Revolution  of  the  Middle  Agea, 
with  such  general  views  of  literature,  society,  and  manners,  as  are  required  to  explain  the  paa- 
»age  from  ancient  to  modern  history.  At  the  head  of  each  chanter  there  is  an  analytical  sum- 
mary, which  will  be  found  of  great  assistance  in  examination  or  in  review.  Instead  of  a  singia 
list  of  sovereigns,  I  have  preferred  giving  full  genealogical  tables,  which  ire  much  clearer  and 
faifinitely  more  satisfactory." 


THE 

FIRST    HISTORY    OF    ROME, 

WITH  aUESTIONS. 

BY    E.    M.    SEWELL, 

Author  of  Amy  Herbert,  &c.,  &c.    One  volume,  16mo.    50  eta. 
Extract  from  Editor's  Preface, 

•'  History  is  the  narrative  of  real  events  in  the  order  and  circumstances  in  which  they  oc 
eurred  ;  and  of  all  histories,  that  of  Rome  comprises  a  series  of  events  more  interesting  and  io 
structive  to  youthful  readers  than  any  other  \hat  has  ever  been  written. 

"  Of  the  manner  in  which  Mrs.  Sewell  has  executed  this  work,  wf.  can  scarcely  speak  in 
terms  of  approbation  too  strong.  Drawing  her  materials  from  the  best — hat  is  :o  say,  the  most 
reliable — sources  she  has  incorporated  them  in  a  narrative  at  once  unostentatious,  perspicuous, 
and  graphic ;  manifestly  aiming  throughout  to  be  cleariy  understood  by  those  for  whom  she 
wrote,  and  to  impress  deeply  and  permanently  on  their  minds  what  she  wrote ;  and  in  botJi  of 
these  aims  we  think  she  has  been  eminently  successful." 

Norfolk  Academy,  Ncr^i  !/t,   Vu. 

I  must  thank  you  for  a  copy  of  "Miss  Sawcll's  Roman  History."  Classical  teachers  have 
long  needed  just  such  a  work :  for  it  is  admitted  by  all  how  essential  to  a  proper  comprehension 
of  the  classics  is  a  knowledge  of  collateral  history.  Yet  most  pupils  are  construing  authors  be- 
fore reaching  an  age  to  put  into  their  hands  the  elaborate  works  we  have  heretofore  had  upoi 
Ancient  History.  Miss  Sewell,  while  she  gives  the  most  important  facts,  has  clothed  them  in  a 
style  at  once  pleasing  and  comprehensible  to  the  most  youthful  mind. 

R.  B.  TSCHUDI, 

Prof,  of  Anc'l  Languages 


THE 

MYTHOLOGY  OF  ANCIENT  GREECE  AND  ITALY, 

FOR  THE  USE  OF  SCHOOLS. 

BY    THOMAS    KEIGHTLEY. 

One  vol.  IGmo.    42  cts. 

"  This  is  a  volume  well  adapted  to  the  purpose  for  which  it  was  prepared.  It  presents,  )•«  » 
very  compendious  anl  convenient  form,  every  thing  relating  to  the  subject,  of  impr  •'.ance  to  tto» 
young  student." 


GENERAL 

HISTORY  OF  CIVILIZATION  IN  EUROPE, 

PROM  THE  FALL  OF  THE  ROMAN  EMPIRE  TO  THE  FRENCH  REVOLUTION. 
BY    IVt.    G  U  I  Z  O  T. 

ighth  American,  from  'he  second  English  edition,  with  occasional  Notes,  by  C.  S.  HENRY,  I).  I> 
One  volume,  12mo.    75  cts. 

*  M.  Guizot,  in  his  instructive  lectures,  has  given  us  an  epitome  of  modem  history,  i!i.«in; 
?uished  by  all  the  merit  which,  in  another  department,  renders  Blackstone  a  subject  of  surf 
f  ecu  liar  and  unbounded  praise.  A  work  closely  condensed,  including  nothing  useless,  uini' 
•1:1?  nothing  essential:  written  with  grace,  and  conceived  and  arranced  with  consummate 
-.hility. "—Boston  Traveller. 

EC5~  Tins  work  is  used  in  Harvard  University,  Unton  College-  University  oj 
Pennsylvania  New-  York  University,  $c.  Sfc. 

\  o 


HISTORICAL 

AND 

MISCELLANEOUS    QUESTIONS. 

BY  RICHMALL   MANGNALL. 

feet  American,  from  the  Eighty-fourth  London  Edition.    With  large  Addition! 

Embracing  the  Elements  of  Mythology,  Astronomy,  Architecture, 

Heraldry,  &c.    Adapted  for  Schoois  n  the  United  States 

BY  MRS.  JULIA   LAWRENCE. 

Illustrated  with  numerous  Engravings.     One  Vobme,  12mo.     $1. 

CONTENTS. 

A  Short  View  of  Sciipture  History,  from  the  Creation  to  the  Return  of  the  Jews — Questioni 
from  the  Early  Ages  to  the  time  of  Julius  Ca3sar—l>;iscellaneous  Questions  in  Grecian  History 
—Miscellaneous  Questions  in  General  History,  ch.^fly  Ancient— Questions  containing  a  Sketcn 
of  the  most  remarkable  Events  from  the  Christian  Era  to  the  close  of  the  Eighteenth  Century— 
Miscellaneous  Questions  in  Roman  History— Questions  in  English  History,  from  the  Invasion  of 
Caesar  to  the  Reformation— Continuation  of  Questions  in  English  History,  from  the  Reformation 
to  the  Present  Time— Abstract  of  Early  British  Ilistorv-  Abstract  of  English  Reigns  from  the 
Conquest— Abstract  of  the  Scottish  Reigns— Abst-act  of  the  French  Rei-rns,  from  Pharamond  to 
Philip  I— Continuation  of  the  French  Reigns,  from  Louis  VI  to  Louis  Phillippe— Questions  Re- 
lating  to  the  History  of  America,  from  its  Discovery  to  riie  Present  Time— Abstract  of  Roman 
Kings  and  ^iost  distinguished  Heroes— Abstract  -^f  the  most  celebrated  Grecians— Of  Heathen 
.Mythology  in  general— Abstract  of  Heathen  Mythology— The  Elements  of  Astronomy— Expla- 
ttoa  of  a  few  Astronomical  Terms— List  of  Constellab'-,e— Questions  on  Common  Subjects— 
•iuestions  on  Architecture— Questions  on  HeraHry— Explanations  of  such  Latin  Words  and 
Phrases  as  are  seldom  Englished — Questions  on  th«  "History  of  the  Middle  Ages. 

"  This  is  an  admirable  work  to  aid  both  teachers  and  parents  in  instructing  children  and  youth, 
uid  there  is  no  work  of  the  kind  that  we  have  seen  iliat  is  so  well  calculated  "  to  awaken  a  spirit 
of  laudable  curiosity  in  young  minds,"  and  to  satisfy  that  curiosity  when  awakened." 


HISTORY  OF  ENGLAND, 
Frour  the  Invasion  of  Julius  Caesar  to  Hie  Reign  of  Queen  Victoria. 

BY    MRS.  MARKHAM. 
A  new  Edition,  with  Questions,  adapted  for  Schools  in  the  United  States. 

BY  ELIZA  ROBBINS, 

Author  of  "American  Popular  Lessons?'  "  Poetry  for  Schools,"  ffc. 

One  Volume,  12mo.     Price  75  cents. 

There  is  nothing  more  needed  in  our  schools  than  good  histories  ;  not  the  dry  compends  b. 
icsent  use,  but  elementary  works  that  shall  suggest  the  moral  uses  of  histoiy,  and  the  provl 
ence  of  God,  manifest  in  the  affairs  of  men. 

Mr.  Markham's  history  was  used  by  that  model  for  all  teachers,  the  late  Dr.  Arnold,  mastei 

-'  the  great  English  school  at  Rugby,  and  agrees  u.  its  character  with  his  enlightened  and  piou* 

>^:W3  of  teaching  history.    It  is  now  several  years  sinc.p  T  adapted  this  history  to  the  form  anC 

ice  acceptable  in  the  schools  in  the  United  States.    I  nave  recently  revised  it,  and  trust  that  i* 

...«y  be  extensively  serviceable  in  education. 

The  principal  alterations  from  the  original  are  a  new  and  more  convenient  division  of  para 
~  phs,  and  entire  omission  of  the  conversations  annexed  to  the  chapters.  In  the  place  of  theat 
I  ive  affixed  questions  to  every  page  that  may  «r  once  facilitate  the  woik  of  the  teacher  and 
'<n  pupil.  The  rational  and  moral  features  of  this  bo<A  first  commended  it  to  me,  and  I  bar* 
>:*!  it  successfully  with  my  own  scholars.  —Extract  fmm  'he  I  w~car>  Editor's  P^'Jax.«, 

12 


A  TREATISE  ON  ALGEBRA. 

FOR  THE  USE  OF  SCHOOLS  AND  COLLEGES. 

•» 
BY  S.  CHASE, 

PROFESSOR  OP  MATHEMATICS  IN  DARTMOUTH  COLLEGE. 

One  volume,  12mo,  S10  pages.    Price  $1. 

•4  Tho  Treatise  which  Prof.  Chase  has  written  for  the  use  of  schools  and  colleges,  seems  to  tw 
to  be  superior  in  not  a  few  respects  to  the  school  Algebras  in  common  use.  The  object  of  the 
jrriter  was,  "to  exhibit  such  a  vie  Y  of  the  principles  of  Algebra,  as  shall  best  prepare  the  stu- 
je.it  for  the  further  pursuit  of  mathematical  studies."  He  has,  we  think,  succeeded  in  this  at- 
tempt. His  book  is  more  complete  in  its  explanations  of  the  principles  of  Algebra  than  any 
text-book  \vith  which  we  are  acquainted.  The  examples  for  practice  are  pertinent,  and  are  suf- 
ficiently numerous  for  the  illustration  of  each  rule. 

'•  Mr.  C.  has  avoided,  by  his  plan,  the  common  fault  of  text  books  on  Algebra — uselessly  mi- 
mer  j»is  sxamples,  and  meagerness  of  explanation  as  respects  the  principles  of  the  science.  The 
ordei  of  treatment  is  judicious.  Mr.  C.  has  added  a  table  offormul&j  for  convenience  of  itfer- 
ence,  in  which  are  brought  into  one  view  the  principles  exhibited  in  different  parts  of  the  book. 
It  will  be  of  great  use  to  the  student.  We  think  the  book  is  well  adapted  to  schools  and  college*, 
into  many  of  which  it  will,  no  doubt,  be  introduced."—  Ch.  Recorder. 


FIRST  LESSONS  IN  GEOMETRY, 

fTPON   THE   MODEL   OF   COLBURN'S    FIRST   LESSONS    IN    ARITHMETIC. 
BY    ALPHEUS    CROSBY, 

PROFESSOR  OF  MATHEMATICS    IN   DARTMOUTH   COLLEGE. 

One  volume,  16mo,  170  pages.    Price  37£  cents. 

This  work  is  approved  of  as  the  best  elementary  trxt-book  on  the  subject,  and  is  very  gene 
rally  adopted  throughout  the  States. 


BURNAM'S  SERIES  OF  ARITHMETICS, 


COMMON  SCHOOLS  AND  ACADEMIES. 

PART  FIRST  is  a  work  on  MENTAL  ARITHMETIC.  The  philosophy  of  the  mode  of  teach*  •$ 
adopted  in  this  work,  is  :  commence  where  the  child  commences,  and  proceed  as  the  child  pi> 
ceeus  :  fall  in  with  hi,?  own  mode  of  arriving  at  truth;  aid  him  to  think  for  himself,  and  do  not 
the  thinking  for  him.  Hence  a  series  of  exercises  are  given,  by  which  the  child  is  made  familiar 
with  the  process,  which  he  has  already  gone  through  with  in  acquiring  his  present  knowledge. 
These  exerci?es  interest  the  child,  and  prepare  him  for  future  rapid  progress.  The  plan  is  ao 
clearly  unfolded  by  illustration  and  example,  that  he  who  follows  it  can  scarcely  fail  to  secure, 
en  the  part  of  his  pupils,  a  thorough  knowledge  of  the  subject.  Price,  20cts. 

PART  SECOND  is  a  work  on  WRITTEN  ARITHMETIC.  It  is  the  result  of  a  long  experience 
in  teaching,  and  contains  sufficient  of  Arithmetic  for  the  practical  business  purposes  of  life.  It 
illustrates  more  fully  and  applies  more  extcndedly  and  practically  the  principle  of  Cancellation 
than  any  other  Arithmetical  treatise.  This  method  as  here  employed  in  connection  with  the  or- 
dinary, furnishes  a  variety  of  illustrations,  which  cannot  fail  to  interest  and  instruct  the  scholar. 
Jt  is  a  prominent  idea  throughout,  to  impress  upon  the  mind  of  the  scholar  the  truth  that  lie  will 
never  discover,  nor  need  a  r/'ov  -rinciple  beyond  the  simple  rules.  The  pupil  is  shown,  by  a 
variety  of  new  modes  of  illustration,  that  new  names  and  new  positions  introduce  no  new  prin- 
ciple, but  that  they  are  merely  matters  of  convenience.  Fractions  are  treated  and  explained  th# 
name  ns  whole  numbers.  Formulas  are  al*=o  given  for  drilling  the  scholar  upon  the  Blacklwrd 
which  will  he  found  of  service  to  m;iny  teachers  nfComttum  Schools.  Price.  50  cw. 

16 


CLASS-BOOK  OF  NATURAL  HISTORY. 


- 


ZOOLOGY 


CES1GNED  TO   AFFORD   PUPILS   IN   COMMON   SCHOOLS   AND    ACADEMIES  « 
KNOWLEDGE  OF  THE  ANIMAL  KINGDOM,  ETC. 

BY  PROFESSOR  J.  J/EGER. 

One  volume,  18mo,  with  numerous  Illustrations.    Price  42  cents. 

"The  distinguished  ability  of  the  author  of  this  work,  both  while  engaged  during  nearly  ten 
years  as  Prof»ssor  of  Botany,  Zoology,  and  Modern  Languages,  in  Princeton  College,  N.  J.,  and 
since  as  a  lecturer  in  some  of  the  most  distinguished  literary  institutions,  together  with  the  rare 
advantages  derived  from  his  extensive  travels  in  various  parts  of  the  world,  under  the  patronage 
of  the  Emperor  of  Russia,  affording  superior  facilities  for  the  acquisition  of  knowledge  in  his 
department,  have  most  happily  adapted  Professor  Jaeger  to  the  task  he  ha£  with  GC  rr.iich  ability 
performed,  viz. :  that  of  presenting  to  the  public  one  of  the  most  simple,  engaging,  and  useful 
Class-Books  of  Zoology  that  we  have  seen.  It  is  peculiarly  adapted  to  the  pinpose  he  had  in 
view,  namely,  of  supplying  a  School  Book  on  this  subject  for  our  Common  Schools  and  Acade- 
mies, which  shall  be  perfectly  comprehensible  to  the  minds  of  beginners.  In  this  respect,  he 
has,  we  think,  most  admirably  succeeded,  and  we  doubt  not  that  this  little  work  will  become  one 
of  the  most  popular  Class  Books  of  Zoology  in  the  country." 

From  Prof.  Tayler  Lewis. 

"  Your  Class-Book  of  Zoology  ought  to  be  introduced  into  all  the  public  and  private  school* 
of  this  city,  and  I  should  rejoice  for  your  own  sake,  and  for  the  sake  of  sound  science,  to  hear  o/ 
its  obtaining  the  public  patronage  which  it  deserves." 

From  Dr.  T.  Romeyn  Beck,  of  Albany. 

"  The  copy  of  your  book  of  which  you  advised  me  last  week,  reached  me  this  morning.  I 
am  pleased  with  its  contents.  Of  its  accuracy  I  can  have  no  question,  knowins  your  long  and 
ardent  devotion  to  the  study  of  Natural  History.  It  will  be  peculiarly  useful  to  the  young  pupil, 
in  introducing  him  to  a  knowledge  of  our  native  animals." 

From  Ret.  Dr.  Campbell,  Albany. 

"Your  'Class-Book'  reached  me  safely,  and  I  am  delighted  with  it;  but  what  is  more  to 
the  purpose,  gentlemen  \*ho  know  something  about  Zoology,  are  delighted  with  it,  such  as  Dr. 
Beck  and  Professor  Cook,  of  our  Academy.  I  have  no  doubt  that  we  shall  introduce  u."' 


PRIMARY  LESSONS  I 

BEING  A  SPELLER  AND  READER,  ON  AN  ORIGINAL  PLAN. 

in  which  one  letter  is  taught  at  a  lesson,  with  its  power;  an  application  being  immsdiatsly 

made,  in  words,  of  each  letter  thus  learned,  and  those  words  being 

•directly  arranged  into  reading  lessons. 

BY  ALBERT  D.  WRIGHT, 

AUTHOR  OP  "ANALYTICAL  ORTHOGRAPHY,"  "PHONOLOGICAL  CHART,"  ETC. 
One  neat  volume,  18mo,  containing  144  pages,  and  28  engravings.    Price  12£  cents,  bound. 


EASY  LESSONS  IN  LANDSCAPE, 

FOR  THE  PENCIL. 
BY  F.  N.  OTIS, 

IN  THREE  PARTS,  EACH  CONTAINING   SIXTEEN   LESSC  NS 

Price  38  cents  each  part. 

These  Lessons  are  intended  for  the  use  of  schools  and  families,  and  are  so  arrangei!  thit  «rflfc 
the  aid  of  the  accompanying  directions,  teachers  unacquainted  with  drawing  may  introduce  1 
«ucce$sfully  into  their  schools;  and  those  unable  to  avail  themselves  of  the  rHv^n.'agea  of  • 
teacher,  may  pursue  the  study  of  drawing  withnm  difficulty. 

17  ^ 


rB  17241 


